ABSTRACTSof the 10th International Conference"Inverse Problems: Modeling and Simulation"
held on May 22 - 28, 2022, Malta

Editors: Alemdar Hasanov Hasanoglu, Roman Novikov, Otmar Scherzer
Cristiana Sebu and Eric Todd Quinto
Managing Editor: Burhan Pektaş
The Conference is organized by The Eurasian Association on Inverse Problems and supported by
the international journals Inverse Problems and Journal of Inverse and Ill-Posed Problems.

ABSTRACTSof the 10th International Conference"Inverse Problems: Modeling and Simulation"
held on May 22 - 28, 2022, in Malta

ABSTRACTS / edited by
A. Hasanov Hasanoglu, R. Novikov, O. Scherzer, C. Sebu, T.Q. Todd
Managing Editor: Burhan Pektaş
ISBN: 978-975-04-0905-9
All rights reserved. Printed in Literatür Yayıncılık, Istanbul, Turkey. No part of this book may be
reproduced, stored, or transmitted without the written permission of Literatür Yayıncılık. For information,please contact with Literatür Yayıncılık, Katip Mustafa Çelebi Mah. İstiklal Cad. Mim Han No:55 Kat:4Beyoğlu-İstanbul, (212) 292 41 20 -literatur@literatur.com.tr

Contents
Contents ....................................................................................................................................................... ii
Preface ....................................................................................................................................................... xiv

Plenary Speakers
Liliana Borcea, Quantitative inverse scattering via reduced order modelling ............................................. 1
Bernd Hofmann, Ill-posedness concepts and the distinguished role of smoothness inregularization for linear and nonlinear inverse problems .............................................................................. 2John C Schotland, Acoustically-modulated electromagnetic inverse source problems .............................. 3
Erkki Somersalo, Hypermodels, sparsity and approximate Bayesian computing ....................................... 4
Gunter Uhlmann, Inverse problems for nonlinear equations ...................................................................... 5
Jun Zou, Direct sampling-type methods for general nonlinear ill-posed inverse problems ........................ 6

Minisymposiums
M1: Recent Advances in Inverse Problems over the Past 20 Years
Commemorative Minisymposium for the 10th IPMS Conference
Jan Boman, Uniqueness and stability questions for inverse Radon Transforms ......................................... 9
Alemdar Hasanov Hasanoglu, Inverse problems for a vibrating Euler-Bernoulli beam based onboundary measured outputs......................................................................................................................... 10
Roman Novikov, Multidimensional inverse scattering problem ............................................................... 11
Otmar Scherzer, Inverse problems of single molecule localization microscopy ...................................... 12Maria-Luisa Rapun, Topological derivative based methods for shape reconstruction ............................ 13
Mikhail Isaev, A simple example of Hölder-logarithmic stability in inverse problems ............................ 14
Karel Van Bockstal, The identification of a time-dependent source in a time-fractional diffusion
equation with non-smooth solutions ........................................................................................................... 15
Alexandre Jollivet, Properties of the Steklov zeta function of a smooth planar domain over thereal axis ....................................................................................................................................................... 16

M2: New Trends in Regularization Theory
Christine Boeckmann, A Modified asymptotical regularization of nonlinear ill-posed problems ........... 18
Yu Deng, On the deautoconvolution problem in the two-dimensional case .............................................. 19ii

Daniel Gerth, A new way of interpreting Tikhonov regularization and its consequence for theestimation of solution smoothness and noise level ..................................................................................... 20
Urve Kangro, On regularized projection methods for ill-posed problems ................................................ 21
Zeljko Kereta, Stochastic gradient descent in Banach Spaces .................................................................. 22
Clemens Meiser, Learned Landweber iteration for the Terahertz Tomography ........................................ 23
Robert Plato, Convergence results for nonlinear Tikhonov regularization with oversmoothing
penalty ......................................................................................................................................................... 24
Toomas Raus, General heuristic rule for choosing regularization parameter in Tikhonov Method .......... 25
Frank Werner, Analysis of (statistical) inverse problems under conditional stability estimates .............. 26
Wensheng Zhang, A mixed regularization method for ill-posed problems............................................... 27
Jorge P. Zubelli, Splitting for jump-diffusion calibration in financial option models............................... 28

M3: Tomographic Inverse Problems
Fulton Gonzalez, Invertible distributions, mean value operators and symmetric spaces .......................... 30
Khalil Hall-Hooper, Estimating hyperparameters in hierarchical Bayesian linear inverse problems ....... 31
Ivan Kazantsev, Algorithms for the Compton scatter imaging in Positron Emission Tomography ......... 32
Leonid Kunyansky, Sparsity-based techniques for hybrid imaging modalities with missing low
frequencies .................................................................................................................................................. 33
Eric Todd Quinto, Novel inverse problems in Compton Tomography ..................................................... 34
Grigory Sabinin, Numerical reconsruction from Fourier Transform on the ball via prolate
spheroidal wave functions ........................................................................................................................... 35
Souvik Roy, Reconstruction of sparse log-conductivity in current density impedance imaging ............... 36
Suman Kumar Sahoo, Symmetry from sectional integrals for convex domains ...................................... 37
Andrei Shurup, Functional-analytical methods in acoustic inverse problems .......................................... 38

M4: Modern Challenges in Inverse Problems Including Boundary Rigidity,
Microlocal Analysis and Cloaking
Minisymposium dedicated to the 70th anniversary of an outstanding expert in inverse problems, Walker
Family Endowed Professor of Mathematics University of Washington, Gunther Uhlmann

Nikolas Eptaminitakis, The solid-fluid transmission problem ................................................................. 40
Allan Greenleaf, Sobolev estimates for multilinear Radon transforms via partition optimization............ 41
Sean Holman, Simultaneous recovery of attenuation and source density in SPECT ................................ 42
Matti Lassas, Inverse problems for finite graphs and cloaking ................................................................. 43
Ru-Yu Lai, Inverse transport and diffusion problems in photoacoustic imaging with nonlinear
absorption .................................................................................................................................................... 44
Hongyu Liu, Wave propagation inside transparent scatterers and applications......................................... 45
Reed Meyerson, Intersection rigidity for simple Riemannian manifolds .................................................. 46
Francois Monard, Mapping properties of X-ray transforms near convex boundaries .............................. 47iii

Benjamin Palacios, Recent developments in Photoacoustic Tomography ................................................ 48
Rakesh, Formally determined inverse problems for hyperbolic PDEs ...................................................... 49
Mikko Salo, Instability mechanisms in inverse problems .......................................................................... 50
Alexandru Tamasan, On the range of the planar X-ray transform of symmetric tensors on the
Fourier lattice of the torus ........................................................................................................................... 51
Jenn-Nan Wang, Estimate the size of an inclusion in a body with complex conductivity using
finite number of measurements ................................................................................................................... 52
Yiran Wang, Integral geometry problems in Lorentzian geometry ........................................................... 53
Ting Zhou, Inverse problems for nonlinear PDEs ..................................................................................... 54

M5: Theoretical and Numerical Results in Geometric Inverse Problems for PDEs
Jone Apraiz, Geometric inverse problems for the Burgers equation and related systems ......................... 56
Jon Asier Bárcena-Petisco, Inverse problems for parabolic equation in networks with loops ................. 57
Larisa Beilina, Adaptive finite element method for solution of inverse and ill-posed problems .............. 58
Umberto Biccari, Multilevel control ......................................................................................................... 59
Carlos Castro, Numerical reconstruction of the conductivity in the 3-D Calderon problem using
the Born approximation............................................................................................................................... 60
Jairo Rocha de Faria, Transcranial direct current simulation: estimation of individually targetedusing the head geometry.............................................................................................................................. 61
Axel Osses, A resistance method for a fluid-obstacle inverse problem ...................................................... 62

M6: Generalized Radon Transforms and Applications
Matthias Beckmann, The Modulo Radon Transforms and its inversion .................................................. 64
Maimouna Bocoum, Structured ultrasound-modulated Optical Tomography .......................................... 65
Aleksander Denisiuk, On range condition of the tensor X-ray transform inn..................................... 67

Joonas Ilmavirta, Ray Transform problems arising from seismology on Mars ........................................ 68
Voichita Maxim, Spectral Compton Camera imaging of polychromatic gamma-ray sources. ................. 69
Mai K. Nguyen-Verger, Some generalized Radon Transforms inspired from imaging technology ......... 70
Daniel Obmann, Augmented NETT regularization of inverse problems .................................................. 71
Stephen Pistorius, Compton scatter tomography: From algebraic and iterative reconstruction tomachine learning ......................................................................................................................................... 72
Michael Quellmalz, Fourier reconstruction in Diffraction Tomography................................................... 73
Jesse Railo, Periodic view on some classical operators associated to Radon Transforms ........................ 74
Lukas Vierus, Diffractive tensor field tomography as an inverse problem for a transport equation ......... 75
Joseph A. O’Sullivan, From analysis to iterative algorithms: A linear systems perspective on the
Broken Ray Transform ................................................................................................................................ 76

iv

James Webber, A joint reconstruction and lambda tomography regularization technique forenergy-resolved X-ray imaging................................................................................................................... 77

M7: Recent Advances in Inverse Problems of Time-Harmonic Wave Propagation
Isaac Harris, Eigenvalue problems for a scatterer with a conductive boundary ....................................... 79
Thorsten Hohage, Uniqueness result for random inverse source problems and applications to
Helioseismic Holograhpy ............................................................................................................................ 80
Andreas Kirsch, Direct and inverse scattering problem for a locally perturbed periodic structure .......... 81
Andreas Kleefeld, Interior transmission eigenvalues for fractal domains ................................................. 82
Shixu Meng, Single mode multi-frequency factorization method for the inverse source problem inacoustic waveguides .................................................................................................................................... 83
Björn Müller, Quantitative passive imaging by iterative helioseismic holography .................................. 84
Roman Novikov, Formulas for phase recovering from phaseless scattering data ..................................... 85
Lukas Pieronek, On complex transmission eigenvalues ........................................................................... 86

M8: Inverse Problems in Science and Engineering
Anwesa Dey, On 3D PET in spherical geometry of data acquisition ......................................................... 88
Pravinkumar Ghodake, Inverse problem for quantification of localized damage using 1D
nonlinear elastic two-wave mixing ............................................................................................................. 89
Yiqian He, Numerical solution of direct/inverse problems with parameter uncertainty of phasechange energy storage wall ......................................................................................................................... 90
Taufiquar Khan, Image reconstruction in Diffuse Optical Tomography: An optimal Bayesianestimator for absorption coefficient ............................................................................................................ 91
Lydie Mpinganzima, An iterative method for the Cauchy problem for the Helmholtz equation ............. 92
Mihaela Pricop-Jeckstadt, Statistical linear inverse problems based on discretely sampled
functional data ............................................................................................................................................. 93
Johannes Schwab, Reconstructing molecular flexibility in Cryogenic Electron Microscopy .................. 94
Cristiana Sebu, Real-time electrical impedance imaging at high AC frequencies .................................... 95

M9: Recent Advances in Analytical and Numerical Methods in Inverse Problems for
Partial Differential Equations
Minisymposium dedicated to the 70th anniversary of a distinguished expert in the field of inverse
problems Professor Michael V. Klibanov

Ugur G. Abdulla, Bang-Bang optimal control in spin dynamics of radical pairs in quantum
biology ........................................................................................................................................................ 97
Maya de Buhan, Carleman-based reconstruction algorithm ..................................................................... 98
Christian Clason, Multiband regularization of a coefficient inverse problem for the wave equation ...... 99v

Anatoly Yagola, Solution of a three-dimensional inverse elastography problem for parametricclasses of inclusions .................................................................................................................................. 100

M10: Computational Methods for Inverse Problems and Applications
Minisymposium dedicated to the 75th anniversary of a distinguished expert in the field of inverse
problems Professor Anatoly Yagola

Gennady Alekseev, Optimization approach in inverse problems of designing thermal cloaking
devices ....................................................................................................................................................... 102
Mikhail Bulatov, Collocation-variation approaches to numerical solution of differential-algebraic
and Volterra equations of the first kind ..................................................................................................... 103
Sergey Buterin, Uniform stability of recovering the Dirac operator with an integral delay from thespectrum .................................................................................................................................................... 104
Vasily Demyanov, Inverse problem challenges in practical subsurface geoengineering applications .... 105
Uno Hämarik, On acceleration of Landweber method for ill-posed problems ....................................... 106
Mikhail V. Klibanov, Carleman estimates for globally convergent numerical methods forcoefficient inverse problems ..................................................................................................................... 107
Marek Kojdecki, On discrepancy principles for Tikhonov regularisation .............................................. 108
Olga Krivorotko, High-performance regularization of multi-parametric inverse problems of
epidemiology and social networks ............................................................................................................ 109
Gulnara Kuramshina, Regularizing methods for predicting bulky molecules vibrational spectra
based on the combined use of AB initio and exparimantal data ............................................................... 110
Maria Kuznetsova, Local solvability and stability of inverse spectral problem for non-self-adjoint
Sturm-Liouville operator ........................................................................................................................... 111
Mikhail Lavrentiev, Fast computation of tsunami wave parameters by measured data inversion ......... 112
Nikolay Nefedov, Asymptotic solutions of inverse coefficient problems for Burger type equations
with interior layer ...................................................................................................................................... 113
Abdugany Satybaev, Finite difference regularized method for solving the generalized onedimensional inverse problem of propagation of the action potential along nervous fiber ........................ 114
Alexander Tikhonravov, Inverse problems in thin film optics: Recent achievements and thenewest trends ............................................................................................................................................. 115
Ye Zhang, On the generalized asymptotical regularization for linear ill-posed problems ....................... 116
Nikolay Zyatkov, Machine learning regularization of inverse problem for Black-Scholes equation...... 117

M11: Inverse Problems for Time-fractional PDEs
Yikan Liu, Inverse source problems for time-fractional diffusion (wave) equations .............................. 119
Walter Simo Tao Lee, A uniﬁed framework for the regularization of ﬁnal value time-fractional
diﬀusion equation ...................................................................................................................................... 120
Lauri Ylinen, Inverse problems for fractional diffusion equation with one measurement ...................... 121
Zhi Zhou, Numerical estimation of a diffusion coefficient in subdiffusion ............................................. 122vi

M12: Electrical Impedance Tomography: Theory and Applications
Elena Beretta, Lipschitz stable determination of polygonal and polyhedral conductivity inclusions
from boundary measurements ................................................................................................................... 124
Valentina Candiani, Machine learning approach for stroke detection in electrical impedance
tomography ............................................................................................................................................... 125
Pedro Caro, The Calderón problem with Lipschitz conductivities ......................................................... 126
Bastian Harrach, Global convergence and stable invertibility for a Robin transmission problem
with finitely many measurements ............................................................................................................. 127
Ivan Pombo, The inverse conductivity problem for complex conductivities with regular jumps ........... 128
Luca Rondi, Interior decay of solutions to elliptic equations .................................................................. 129
Silvia Sciutto, Generative models for Electrical Impedance Tomography .............................................. 130

M13: Inverse Problems in Geomathematics and Seismology
Eric Bonnetier, A stability estimate for an inverse electroseismic problem............................................ 132
Ioan Ionescu, Earthquake nucleation: direct and inverse problems ......................................................... 133
Kundan Kumar, Coupled flow and geomechanics in fractured porous medium .................................... 134
Matti Lassas, Deep learning architectures for nonlinear operator functions and inverse problems
for wave equations .................................................................................................................................... 135
Laurent Seppecher, Regularization for seismic sources inversion from interferometric data ................ 136
Darko Volkov, Stability properties for a class of inverse problems with applications to neuralnetwork solutions ...................................................................................................................................... 137
Gertjan van Zwieten, Bayesian inference of seismic events from local deformations using the
Weakly-enforced Slip Method .................................................................................................................. 138

M14: Mathematical Methods in Tomography Across the Scales
Andrea Aspri, Data driven regularization ............................................................................................... 140
Florian Faucher, Quantitative seismic imaging using reciprocity-based methods enablingarbitrary probing sources........................................................................................................................... 141
Leon Frischauf, Data-driven methods in inverse problems ..................................................................... 142
Simon Hubmer, A frame decomposition of the atmospheric tomography operator................................ 143
Tim Jahn, A Probabilistic Oracle inequality and quantification of uncertainty of a modified
discrepancy principle for statistical inverse problems .............................................................................. 144
Kamran Sadiq, A Fourier approach to the inverse source problem in an absorbing and scattering
medium with applications to Optical Molecular Imaging ......................................................................... 145

vii

Ekaterina Sherina, Displacement field estimation utilizing speckle information for parameterrecovery in quantitative elastography ....................................................................................................... 146
Leopold Veselka, Quantitative OCT reconstructions for dispersive media ............................................. 147

M15: Inverse Source Problems with Applications to Planetary Sciences and Medical
ImagingPaul Asensio, Inverse problem of source identification in electroencephalography (EEG) .................... 149
Christian Gerhards, Some inverse magnetization problems in Geoscience........................................... 150
Cristóbal Villalobos Guillen, Some measure-theoretic aspects of planar magnetization
reconstruction ............................................................................................................................................ 151
Xinpeng Huang, Relating hardy components of compactly supported magnetizations on sphere .......... 152
Alexander Kegeles, A “simple” approach to the magneto-static inverse problem .................................. 153
Masimba Nemaire, A layer potential approach to functional and clinical brain imaging ....................... 154

M16: Recent Advances in Inverse Problems and Distributed Parameter Systems
Minisymposium in honor of Professor H. T. Banks' long career in applied mathematics and hisleadership in inverse problems

Azmy Ackleh, Finite difference schemes for a structured population model in the space ofmeasures .................................................................................................................................................... 156
John E. Banks, Population dynamics in applied ecology: models & experiments .................................. 157
John A. Burns, Numerical methods for control, optimization, design and estimation of infinite
dimensional systems.................................................................................................................................. 158
Daniel J. Inman, Why are there so many structural health monitoring algorithms ................................. 159
Fumio Kojima, Inverse problem for electromagnetic propagation in human muscle tissues using
hamilton monte carlo method.................................................................................................................... 160

M17: Parameter Identification Problems for PDEs: Theoretical and Computational
Aspects
Sarah Eberle, Reconstruction of Lamé parameters in linear elasticity.................................................... 162
Thanh Trung Nguyen, Model and source identification problems for a system of advectionreaction equations and applications in water quality................................................................................. 163
Tran Nhan Tam Quyen, Electrical impedance tomography with partial Cauchy data ........................... 164
Francesco Attilio Bruno Silva, A Reduced Basis Ensemble Kalman Method for Inverse Problems ...........
Irwin Yousept, Acoustic full-waveform inversion via optimal control ................................................... 165

M18: Coefficient Identification ProblemsviiiNatalia Bondarenko, An inverse problem for the Sturm-Liouville equation with analyticaldependence on the eigenparameter in the boundary condition ................................................................. 167
Nataliia Kinash, Inverse problems to reconstruct space- and time-dependent sources in evolution
equations containing fractional Laplace operators .................................................................................... 168
Daniel Lesnic, Recovery of a space-dependent rate of reaction in a thermal-wave model of bioheat transfer ............................................................................................................................................... 169
Anthony Mulholland, Identification of heterogeneous material coefficients using ultrasonic arrays .... 170
Bolatbek Rysbaiuly, An iteration method for solving the inverse problem of freezing soil ................... 171
Marian Slodicka, Uniqueness for an inverse source problem of determining a space dependentsource in a non-autonomous parabolic equation ....................................................................................... 172

M19: Modern Challenges in Imaging, Tomography, and Radon Transforms
Minisymposium dedicated to the 70th anniversary of an outstanding expert in inverse problems andimaging sciences, Professor Todd Quinto

Anuj Abhishek, An adaptive nonparametric estimator in an inverse problem for Exponential
Radon Transform ...................................................................................................................................... 174
Mark Agranovsky, Domains with algebraic X-ray transform ................................................................ 175
Raluca Felea, Microlocal analysis of the crosswell and the walkaway seismic data ............................... 176
Jürgen Frikel, Microlocal analysis in tomographic reconstructions ....................................................... 177
Alexander Katsevich, Resolution of 2D reconstruction of functions with nonsmooth edges fromdiscrete Radon transform data ................................................................................................................... 178
Peter Maass, Regularization by architecture: Learning with few data and applications to CT ............... 179
Ronny Ramlau, Regularized recycling methods for linear inverse problems with applications toadaptive optics........................................................................................................................................... 180
Otmar Scherzer, Quantitative inverse problems in visco-acoustic media and evaluation of
attenuation model uncertainties................................................................................................................. 181

M20: Inverse Obstacle and Control Problems in Mechanics
Onur Baysal, Numerical method for source identfication problem related to dynamical Kirchhoff
plate equation ............................................................................................................................................ 183
Carlos Borges, High resolution inverse obstacle scattering using multiple frequency data .................... 184
Peter Elbau, An inverse obstacle problem for the time-dependent heat equation ................................... 185
Andreas S. Hauptmann, Implementation of the enclosure method for some inverse crack
problems .................................................................................................................................................... 186
Alexandre Kawano, Uniqueness in load identification in vibrating nanoplates ..................................... 187
Victor A. Kovtunenko, Mathematical model of crack diagnosis ............................................................ 188
Yulia M. Meshkova, On homogenization of periodic hyperbolic systems.............................................. 189
Virginia Selgas, Steklov and modified transmission eigenvalues as target signatures in an inverse
fluid-solid interaction problem .................................................................................................................. 190

ix

Yuliya E. Spivak, Optimization approach in 2D problems of static fields cloaking ............................... 191
Ping Wu, Elastoplastic modelling and experiments for inversel problems based on entropy
approaches ................................................................................................................................................. 192

M21: Imaging Modalities: Recent Advances and Beyond
Melody Alsaker, Ultrasound data as a prior in Ttoracic imaging with Electrical Impedance
Tomography .............................................................................................................................................. 194
Andrea Ebner, Regularization of inverse problems by filtered diagonal frame decomposition ............. 195
Tatiana Bubba, Simultaneous reconstruction of emission and attenuation in passive gamma
emission tomography of spent nuclear fuel............................................................................................... 196
Tobias Kluth, Deep image prior reconstruction for 3D magnetic particle imaging ................................ 197
Volker Michel, Geophysical and medical imaging: What they can learn from each other...................... 198

M22: Inverse Problems with Data-Driven Methods and Deep Learning
Vegard Antun, Still no free lunch – On AI generated hallucinations and the accuracy-stability
trade-off in inverse problems .................................................................................................................... 200
Riccardo Barbano, A Bayesian deep image prior ................................................................................... 201
Sören Dittmer, Ground truth free denoising by optimal transport........................................................... 202
Margaret Duff, Regularising inverse problems with generative machine learning models .................... 203
Samy Wu Fung, Efficient training of infinite-depth neural networks via Jacobian-Free
Backpropagation ....................................................................................................................................... 204
Reinhard Heckel, Measuring and enhancing robustness in deep learning based compressive
sensing ....................................................................................................................................................... 205
Johannes Hertrich, Stochastic normalizing flows for inverse problems: A Markov chains
viewpoint ................................................................................................................................................... 206
Peter Jung, Solving MMV problems via algorithm unfolding ............................................................... 207
Ulugbek Kamilov, Recovery analysis for plug-and-play priors using the restricted eigenvalue
condition.................................................................................................................................................... 208
Maximilian Kiss, 2DeteCT: A large 2D expandable, trainable, experimental computed
tomography data collection for machine learning ..................................................................................... 209
Jan Macdonald, Inverse problems with deep neural networks – robustness included? .......................... 210
Subhadip Mukherjee, Data-driven adversarial regularization for imaging inverse problems ................ 211
Jenni Poimala, Learned speed of sound correction for photoacoustic tomography ................................ 212
Tom Tirer, Solving ill-posed inverse problem with pretraned denoisers, gans and super-resolvers:
The BP term and the correction filter ........................................................................................................ 213xM23: Theory and Numerics for Inversion Strategies
Takashi Furuya, Local recovery of a piecewise constant anisotropic conductivity in EIT on
domains with exposed corners .................................................................................................................. 215
Durga Prasad Challa, Extraction of the mass density by embedding contrasted small inclusions ........ 216
Shumin Li, Carleman estimates and inverse problems for the coupled quantitative thermo-acoustic
equations ................................................................................................................................................... 217
Manabu Machida, Diffuse optical tomography with a simulated annealing Monte Carlo algorithm..... 218
Gen Nakamura, Hokkaido Uniqueness for the inverse boundary value problem of piecewise
homogeneous anisotropic elasticity in the time domain ........................................................................... 219
Hiroshi Takase, Inverse source problems for wave equations on Lorentzian manifolds ........................ 220
Zhidong Zhang, On the identification of source term in the heat equation from sparse data.................. 221

M24: Inverse Problems via Topological Derivatives
Ana Carpio, Topological derivative based Bayesian inference for inverse scattering problems ............ 223
Yuri Flores-Alburquerque, Reconstruction of sharp interfaces in time-domain full waveform
inversion .................................................................................................................................................... 224
Bochra Mejri, Topological sensitivity analysis for identification of voids under Navier’s boundary
conditions in linear elasticity .................................................................................................................... 225
Luca Ratti, Detection of small cardiac ischemic regions from boundary measurement via
topological gradient ................................................................................................................................... 226
Won-Kwang Park, A real-time identification of small conductivity inhomogeneity via topologicalderivative ................................................................................................................................................... 227
Manuel Pena, Topological derivative algorithm tested against experimental data: The threedimensional Fresnel database .................................................................................................................... 228

M25: Inverse Problems in Biomedical and Material Imaging
Elie Bretin, Direct inversion method for quasi-static medical elastography: Stability anddiscretization ............................................................................................................................................. 230
Elisabeth Brusseau, Mapping the relative shear modulus within biological tissues from internal
displacement fields measured in quasi-static ultrasound elastography ..................................................... 231
Luca Calatroni, Recent advances in correlation-based super-resolution fluorescent microscopy:
From sparse/non-convex to generative approaches................................................................................... 232
Damiana Lazzaro, A deep convolutional neural network for brightness estimation in fluorescenceimaging...................................................................................................................................................... 233
Serena Morigi, Spatially adaptive image reconstruction in Electrical Impedance Tomography............. 234
Angèle Niclas, High sensitivity imaging of defects in elastic waveguides using near resonancefrequencies ................................................................................................................................................ 235

xiElena Loli Piccolomini, Rising: A new framework for few-view tomographic image
reconstruction with deep learning ............................................................................................................. 236
Simone Rebegoldi, A scaled adaptive FISTA-like algorithm for super-resolution image
microscopy ................................................................................................................................................ 237
Gabriele Scrivanti, A variational approach for joint image recovery-segmentation based on
spatially varying generalised Gaussian models ......................................................................................... 238
Elisabetta Vallarino, Optimal regularized estimation of the cross-power spectrum from indirect
measurements: Theoretical results and application to brain connectivity ................................................. 239
Alessandro Viani, Free hyper-parameter selection and averaging in Magneto/Electroencephalography inverse problem............................................................................................................. 240

M26: Variational Methods for Inverse Problems in Imaging
Robert Beinert, Robust pca via regularized reaper and matrix-free proximal algorithms ...................... 242
Benjamin Berkels, Joint exit wave reconstruction and image registration as a least-squares
problem ..................................................................................................................................................... 243
Kristian Bredies, Optimal-transport-based approaches for dynamic image reconstruction .................... 244
Tatiana Bubba, Deep neural networks for inverse problems with pseudodifferential operators: An
application to Limited Angle Tomography ............................................................................................... 245
Vincent Duval, Towards off-the-grid algorithms for total variation regularized inverse problems ........ 246
Richard Huber, Pixel-Driven Tomography: Analysis and the Gratopy Toolbox ................................... 247
Dirk Lorenz, Steklov and modified transmission eigenvalues as target signatures in an inverse
fluid-solid interaction problem .................................................................................................................. 248
Jean-Christophe Pesquet, Penalized methods for solving constrained variational problems inimage recovery .......................................................................................................................................... 249
Rudolf Stollberger, Variational methods for functional and quantitative MRI....................................... 250
Robert Tovey, Accelerating the solution of sparse dynamic inverse problems using tools fromdynamical programming ........................................................................................................................... 251

M27: Inverse Problems in Scattering Theory and Geometry
Tracey Balehowsky, Determining a Lorentzian metric from the source-to-solution map for the
relativistic Boltzmann equation................................................................................................................. 253
Jinpeng Lu, Stability of Gel’fand’s inverse boundary spectral problem via the unique continuation ..... 254
Leyter Potenciano Machado, The fixed angle scattering problem with a first-order perturbation ........ 255
Petri Ola, Simultaneous recostruction of conductivity, boundary shape and contact impedances in
EIT ............................................................................................................................................................ 256
Teemu Tyni, Inverse scattering problems for the biharmonic operator ................................................... 257

M28: Regularization Methods and Applications in Statistics and Econometricsxii

Fredrik Hildrum, Total variation-based Lavrentiev regularisation of monotone problems ................... 259
Mirza Karamehmedovic, Localization of moving sources: Uniqueness, stability, and Bayesianinference .................................................................................................................................................... 260
Pierre Maréchal, On the deconvolution of random variables ................................................................. 261
Clément Marteau, Supermix: Sparse regularization for mixtures .......................................................... 262
Elena Resmerita, On Hamilton-Jacobi PDEs and image denoising models with certain nonadditive noise ............................................................................................................................................ 263
Tuomo Valkonen, Regularisation, optimisation, subregularity ............................................................... 264
Anne Vanhems, A mollifier approach to the nonparametric instrumental regression problem ............... 265

Poster Session
Sonia Foschiatti, Stability for a special class of anisotropic conductivities via an ad-hoc misfit
functional .................................................................................................................................................. 267
Hjørdis Amanda Schlüter, Jacobian of solutions to the conductivity equation in limited view ............ 268
Huynh Phuoc Truong, Optimal experimental design for sound source localization .............................. 269

xiii

Preface
The Tenth International Conference "Inverse Problems: Modeling and Simulation" is held during
May 22 – 28, 2022, in the Paradise Bay Resort Hotel, Malta, under the auspices of The Eurasian
Association on Inverse Problems (EAIP), and also the leading international journals Inverse
Problems and Journal of Inverse and Ill-Posed Problems.
As with the previous nine conferences in this series, the main objective of this conference is to be
multidisciplinary and international, by bringing together scientists working on various topics of
inverse problems in diverse areas, such as mathematics, statistics, engineering, economics, finance,
physics, chemistry, biology, medicine, meteorology and computer science.
The conference brings together more than 200 internationally known experts on inverse problems,and exhibitors from over 30 countries world-wide.
The conference program includes 6 plenary lectures as well as invited lectures given in theframework of 28 minisymposiums. The topics of the conference go through advances in inverse
and ill-posed problems over the past 20 years as well as emerging methods in data science.
We hope that this book of abstracts will be useful to those who are interested in inverse and illposed problems: theory, numerical implementations and applications.
We would like to thank Mr. Kenan Kocatürk, founder and manager of the publishing house
Literatür, for his support.

xiv

PLENARY SPEAKERS

xv

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

QUANTITATIVE INVERSE SCATTERING VIA REDUCED ORDER MODELING

Liliana Borcea
Department of Mathematics, Universit of Michigan, USA, borcea@umich.edu

Abstract
We will discuss an inverse problem for the wave equation, where an array of sensors probes an unknown
heterogeneous medium with waves and measures the echoes. The goal is to determine scattering structures in themedium modeled by a reflectivity function.
Much of the existing imaging methodology is based on a linear least squares data fit approach. However, themapping between the reflectivity and the wave measured at the array is nonlinear and the resulting images haveartifacts. We will show how to use a reduced order model (ROM) approach to solve the inverse scattering problem. The
ROM is data driven i.e., it is constructed from the data, with no knowledge of the medium. It approximates the wave
propagator, which is the operator that maps the wave from one-time step to the next. We will show how to use the
ROM to:(1) Remove the multiple scattering (nonlinear) effects from the data, which can then be used with any
linearized inversion algorithm.
(2) Obtain a well-conditioned quantitative inversion algorithm for estimating the reflectivity

1

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

ILL-POSEDNESS CONCEPTS AND DISTINGUISHED ROLE OF SMOOTHNESS IN
REGULARIZATION FOR LINEAR AND NONLINEAR INVERSE PROBLEMS

Bernd Hofmann
Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany
E-mail: hofmannb@mathematik.tu-chemnitz.de

Abstract
In a first part, we consider different concepts of ill-posedness for mathematical models of inverse problems
expressed by linear and non-linear operator equations in infinite dimensional Hilbert and Banach spaces. The conceptsof Hadamard and Nashed are recalled, which are appropriate for linear operator equations. Compact linear forward
operators in Hilbert spaces allow for a pre-selection of the degree of ill-posedness by verifying the decay rate of thesingular values indicating the operator smoothness. The interplay of operator smoothness and solution smoothness is
responsible for opportunities and limitations of stable approximate solutions to inverse problems by regularization. For
non-linear operator equations, the nature of ill-posedness may vary with the solution point. The presented concept of
local ill-posedness is adapted to that feature.
In a second part, the impact of different varieties of smoothness on variants of variational regularization isunder consideration. In this context, variational source conditions represent a sophisticated tool for expressing thesolution smoothness with respect to the character of the forward operator, for non-linear problems even in combination
with the occurring structure of non-linearity. Also the role of conditional stability estimates in combination with
Tikhonov regularization is outlined. Several example situations for applying regularization approaches are presented,
including sparsity-promoting versions. We also discuss the specific difficulties and some recent results for the
Tikhonov regularization with oversmoothing penalties.
This research is supported by the German Research Foundation (DFG) under grant HO 1454/12-1 andembedded in the Austrian/German joint research project “Novel Error Measures and Source Conditions ofRegularization Methods for Inverse Problems (SCIP)” with Prof. Otmar Scherzer (Vienna) based on the D-A-CH LeadAgency Agreement.

2

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

ACOUSTICALLY MODULATED ELECTROMAGNETIC INVERSE SOURCE
PROBLEMSJohn C Schotland
Department of Mathematics, Yale University, USA, john.schotland@yale.edu

Abstract
The inverse source problem for the Maxwell equations is of fundamental interest and considerable
practical importance, with applications ranging from geophysics to biomedical imaging. The problem is usually statedin the following form: determine the electric current density from boundary measurements of the electric and magnetic
fields. It is well known that this problem is underdetermined and does not admit a unique solution.
In this work we propose an alternative approach to the electromagnetic inverse source problem. In thisapproach the electric current density as well as the conductivity, electric permittivity and magnetic permeability arespatially modulated by an acoustic wave. In this manner, we find that it is possible to uniquely recover the current
density from boundary measurements of the fields with Lipschitz stability.
Numerical simulations are used to illustrate the analytical results.

3

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

HYPERMODELS, SPARSITY AND APPROXIMATE BAYESIAN COMPUTING

Erkki Somersalo
Department of Mathematics, Applied Mathematics and Statistics
Case Western Reserve University, USA, ejs49@case.edu

Abstract
In numerous applications involving an underdetermined large scale inverse problem, sparsity of the solution isa desired property. In the Bayesian framework of inverse problems, sparsity requirement of the solution may be
implemented by properly defining the prior distribution. While Gaussian priors are not well suited for promoting
sparsity, certain hierarchical, conditionally Gaussian models have been demonstrated to be efficient. In this talk, we
review a general class of conditionally Gaussian hypermodels that provide a flexible framework for promoting sparsityof the solution, and discuss approximate iterative methods that can be used both for finding single sparse solutions aswell as for approximate sampling of the posterior distributions.
The motivation for this work comes from an ongoing work on brain imaging using magnetoencephalography
data.

4

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

INVERSE PROBLEMS FOR NONLINEAR EQUATONS

Gunter Uhlmann
Department of Mathematics, University of Washington, USA, gunther@math.washington.edu;
Institute for Advanced Study, HKUST, Hong Kong

Abstract
We will survey recent developments in the solution of inverse problems for nonlinear equations. The
nonlinearity helps to solve several inverse problems that cannot be solved for the linearized equations.

5

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

DIRECT SAMPLING-TYPE METHODS FOR GENERAL NONLINEAR ILL-POSEDINVERSE PROBLEMS
Jun Zou
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China, zou@math.cuhk.edu.hk

Abstract
In this talk we will have an overall review of the systematic developments of direct sampling methods (DSMs)
for solving general nonlinear inverse problems of PDEs. DSMs were initially proposed for solving inverse acoustic
scattering problems with either far-field or near-field data [7][8], then developed for inverse electromagnetic scattering
problems [6], and further extended for solving several representative non-wave type inverse problems, including
electric impedance tomography[5], diffusive optical tomography [3], inversion of Radon transform [1], as well as
recovering moving inhomogeneous inclusions [4]. DSMs have also been proved recently to be applicable to thesimultaneous reconstruction of inhomogeneous inclusions of different physical nature [2]. The DSMs arecomputationally cheap, highly parallel, and robust against noise, particularly applicable to the cases when very limited
data is available. General motivations, principles and justifications of DSMs are presented in this talk. Numericalexperiments are demonstrated for various inverse problems.
There are quite intensive studies of DSMs, and some of the references for those most representative inverse
problems are listed below.
These research projects were supported by Hong Kong RGC General Research Fund (Projects 14306921 and
14306719).

References
1. Yat Tin Chow, Fuqun Han and Jun Zou, A direct sampling method for the inversion of the Radon transform, SIAM J.
Imaging Sci. 14, 1004-1038 (2021).
2. Yat Tin Chow, Fuqun Han and Jun Zou, A direct sampling method for simultaneously recovering inhomogeneous
inclusions of different nature, SIAM J. Sci. Comput., 43, A2161-A2189 (2021).
3. Yat Tin Chow, Kazufumi Ito, Keji Liu and Jun Zou, Direct sampling method for diffusive optical tomography, SIAM
J. Sci. Comput. 37, A1658-A1684 (2015).
4. Yat Tin Chow, Kazufumi Ito and Jun Zou, A time-dependent direct sampling method for recovering moving
potentials in a heat equation, SIAM J. Sci. Comput. 40, A2720-A2748 (2018).
5. Yat Tin Chow, Kazufumi Ito and Jun Zou, A direct sampling method for electrical impedance tomography, Inverse
Problems 30, 095003 (2014).
6. Kazufumi Ito, Bangti Jin and Jun Zou, A direct sampling method for inverse electromagnetic medium scattering,
Inverse Problems 29, 095018 (2013).
7. Kazufumi Ito, Bangti Jin and Jun Zou, A direct sampling method to an inverse medium scattering problem, Inverse
Problems, 28, 025003 (2012).
8. Roland Potthast, A study on orthogonality sampling, Inverse Problems, 26, 074015 (2010).

6

MINISYMPOSIUMS

7

MINISYMPOSIUM
M1: Recent Advances in Inverse Problems over the Past 20 Years
Commemorative Minisymposium for the 10th IPMS Conference
Organizers:
Alemdar Hasanov Hasanoglu, Kocaeli University, Turkey, alemdar.hasanoglu@gmail.com
Eric Todd Quinto, Tufts University, USA, Todd.Quinto@tufts.edu
The First IPMS Conference was organized by John R. Cannon, Alemdar Hasanov, Zuhair M. Nashed, Albert
Tarantola, Alex Tolstoy and Fadil Santosa, in 2002, in Öludeniz - Fethiye, Turkey. Since then, thesemeetings have occurred biennially. The following traditional features of the conference have been important
aspects of the conferences from the beginning: friendship, collaboration and support of young scientists;celebrating the anniversaries of our outstanding colleagues; rewarding young scientists in order to encouragethem; rewarding our distinguished colleagues The EAIP (Eurasian Association on Inverse Problems) Award
to recognize their outstanding scientific contributions to the field of inverse problems and continuous efforts
to foster cooperation between researchers of Eurasian countries. Interest in the IPMS conferences has beensteadily growing over the last decade. Every effort has been made to maintain those traditional features of
IPMS Conferences, and this has made these conferences unique and exciting, starting with the inaugural
meeting in 2002.
This minisymposium consists of topical survey lectures by distinguished experts on inverse problems
who have played an extraordinary role in establishing the tradition of the IPMS conference series as well as
lectures from EAIP Award and EAIP Young Scientist Award winners.
We extend our thanks to all our excellent keynote and minisymposium speakers, conference attendees,
students, exhibitors and guests who have made these IPMS conferences most successful and exciting events.

8

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

UNIQUENESS AND STABILITY QUESTIONS FOR INVERSE RADON TRANSFORMS
Jan Boman
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden, jabo@math.su.se

Abstract
In Computerized Tomography it is often desirable to reconstruct a function f (x) on a proper subset S of the
support of f , the region of interest, from measurements of the Radon transform Rf (L) on the set of lines L thatintersect S . Well known examples show that this is not possible in general. Instead it was believed for a while that (indimension 2) it was necessary to know the full Radon transform Rf for computing f in a region of interest.
However, beginning around 2000 new methods were invented that made it possible in some cases to reconstruct f ina region of interest based on considerably less than complete data, [1]. For instance, if J is a segment of a line L0
that contains L0  supp f , and Rf (L) is known for all lines L that intersect J , then one can compute the(one-dimensional) Hilbert transform of the restriction of f to J . This has led to much recent work on the Region ofInterest problem, for instance to study of uniqueness and stability for inverting the so-called truncated Hilberttransform.
In SPECT one measures the weighted Radon transform ( , x)  exp   0 a ( x  t I ) dt Already in the 1980s inversion of R  was performed in hospitals, although it was not known if R  was injective on
compactly supported f for arbitrary attenuation function a(x) . In 2002 Roman Novikov solved this problem bygiving an explicit formula for the solution f of the equation R f  g , [2].

References
1. J. Edward, Pre-compactness of isospectral sets for the Neumann operator on planar domains, 1. R. Clackdoyle and
M. Defrise, Tomographic reconstruction in the 21st century, IEEE Signal Proc. Mag. 27, 60-80 (2010).
2. R. Novikov, An inversion formula for the attenuated X-ray transformation, Ark. Mat. 40, 145-167 (2002).

9

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

INVERSE PROBLEMS FOR A VIBRATNG EULER-BERNOULLI BEAM
BASED ON BOUNDARY MEASURED OUTPUTS

Alemdar Hasanov Hasanoglu
Kocaeli University, Turkey, alemdar.hasanoglu@gmail.com

Abstract
In this talk, we will describe inverse problems based on novel mathematical models emerging from important
engineering applications. Based on unified approach, we will analyze the distincyive properties of the following three
types of inverse problems: inverse source problems, inverse boundary value problems and inverse coefficient problems
formulated for the variable coefficients dynamic Euler-Bernoulli equationm( x)utt   ( x)utt  r ( x)uxx  F ( x, t ) ( x, t )  T : ( x, t )  R 2 : x  (0, l ), t  (0, T ) .



We will outline reconstruction methods and some benchmark examples to illustrate our results.
These inverse problems are based on the following types available boundary measured outputs: measured output:
the measured deflection, slope or moment [1-5]. The proposed approach is based on a weak solution theory for PDEs,
Tikhonov regularization combined with the adjoint method [5]. The Neumann-to-Dirichlet, as well as Neumann-toNeumann operators corresponding to the inverse coefficient problems, are introduced. It is shown that these operators
are compact and Lipschitz continuous. The last property allows us to prove the existence of a quasi-solution of theconsidered inverse problems. Frechet dierentiability of the corresponding Tikhonov functionals are proved and explicit
formulas for the Frechet gradients are derived by making use of the unique solutions of the corresponding adjoint
problems.
Numerical examples with random noisy measured outputs are presented to illustrate the validity andeffectiveness of the proposed approach.

References
1. A. Hasanov and O. Baysal, Identification of unknown temporal and spatial load distributions in a vibrating EulerBernoulli beam from Dirichlet boundary measured data, Automatica, 71 (2016), 106—117.
2. A. Hasanov, O. Baysal and C. Sebu, Identification of an unknown shear force in the Euler-Bernoulli cantilever beamfrom measured boundary deflection, Inverse Problems 35(5) (2019) 115008. DOI: 10.1088/1361-6420/ab2a3.
3. A. Hasanov and O. Baysal, Identification of a temporal in a cantilever beam from measured boundary bending
moment,Inverse Problems 35(10) (2019) 105005.. DOI: 10.1088/1361-6420/ab2aa9.
4. A. Hasanov, Vladimir Romanov and Onur Baysal, Unique recovery of unknown spatial load in damped EulerBernoulli beam equation from final time measured output, Inverse Problems 37(2021) 075005. DOI:10.1088/13616420/ac01fb
5. A. Hasanov Hasanoglu and V.G. Romanov, Introduction to Inverse Problems for Differential Equations, 2nd Edn.,
Springer, New York, 2021

10

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

Roman G. Novikov
CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France; IEPT RAS,117997
Moscow, Russia
E-mail: novikov@cmap.polytechnique.fr

Abstract
We give a short review of old and recent results on the multidimensional inverse scattering problem for the
Schrödinger equation. A special attention is paid to approximate but mathematically justified and numerically efficientreconstructions of the potential from scattering data which can be measured in practice.
In particular, our considerations include reconstructions from monochromatic scattering data which arenonoverdetermined or/and phaseless. Potential applications include acoustic tomography, tomographies using
elementary particles and phaseless inverse X-ray scattering.
This talk is based, in particular, on the works [1- 5]

References[1] A.D. Agaltsov, T. Hohage, R.G. Novikov, An iterative approach to monochromatic phaseless inverse scattering,
Inverse Problems 35(2), 024001 (34pp) (2019)[2] R.G. Novikov, Approximate inverse quantum scattering at fixed energy in dimension 2, Proc. Steklov Math. Inst.
225, 285-302 (1999)[3] R.G. Novikov, The d-bar approach to monochromatic inverse scattering in three dimensions, J.Geom. Anal. 18,
612-631 (2008)[4] R.G. Novikov, An iterative approach to non-overdetermined inverse scattering at fixed energy, Sbornik:
Mathematics 206(1), 120-134 (2015)[5] R.G. Novikov, Inverse scattering without phase information, Seminaire Laurent Schwartz - EDP et applications(2014-2015), Exp. No16, 13p.

11

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

INVERSE PROBLEMS OF SINGLE MOLECULE LOCALIZATION MICROSCOPY

Otmar Scherzer
University of Vienna, Austria, otmar.scherzer@univie.ac.at

Abstract
Single molecule localization microscopy (SMLM) is a high resolution imaging technique, which allows tovisualize single molecules. The basic principle consists in chemically attaching fluorescent dyes to the molecules,
which after excitation with a strong laser light emit light. In this paper we derive some models describing the
propagation of light in SMLM experiments via Maxwell's equations. This is the basis of formulating inverse problems
related to SMLM. We show that the current status of practical reconstruction can be derived as a simplified solution of
the general inverse problem.
This is joint work with Montse Lopez Martinez (TU Vienna), Kamran Sadiq (RICAM, Linz), Magdalena
Schneider (TU Vienna), John Schotland (University of Michigan), Gerhard Schuetz (TU Vienna).

12

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

TOPOLOGICAL DERIVATIVE BASED METHODS FOR SHAPE RECONSTRUCTION
Maria-Luisa Rapún
Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, ETSI Aeronáutica y del Espacio, Universidad
Politécnica de Madrid, 28040 Madrid, Spain, marialuisa.rapun@upm.es

Abstract
Solving inverse problems related with the detection of multiple objects and the reconstrucion of their shapes
when no a priori information about their number, size or location is provided, is an important field in Applied
Mathematics and Physics that arises in a large number of different industrial and engineering applications such as nondestructive testing, geophysical exploration, biomedicine, radar imaging and antenna design.
In this work we propose numerical methods to solve inverse obstacle problems by topological derivative based
methods. The topological derivative of a shape functional measures the sensitivity of such functional to having an
infinitesimal object at each point of the region of interest. The idea behind these methods is to generate an indicatorfunction able to classify each point as either belonging to the background medium or to an object, without any a priori
assumption. We will show closed-form formulae of this derivative for several model problems, which will involve thesolution to a forward problem and to a related adjoint one.
The performance of these methods will be illustrated throughout a wide gallery of numerical experiments
covering applications that include inverse scattering problems in 2D and 3D acoustics [1,2,3], electromagnetism [4,5],
holographic 3D imaging [6] and inverse heat conduction problems [7].

References
1. A. Carpio and M.L. Rapún, Solving inhomogeneous inverse problems by topological derivative methods, Inv. Prob.,
24, art. 045014 (2008).
2. J.F. Funes, J.M. Perales, M.L. Rapún and J.M. Vega. Defect detection from multi-frequency limited data via
topological sensitivity, J. Math. Imaging Vision, 55, 19-35 (2016).
3. F. Le Louër and M.L. Rapún. Detection of multiple impedance obstacles by non-interative topological gradiend
based methods, J. Comput. Phys, 388, 534-560 (2019)
4. F. Le Louër and M.L. Rapún, Topological derivatives for solving inverse multiple scattering problems in threedimensional electromagnetism. Part I: One step method, SIAM J. Imaging Sci., 10, 1291-1321 (2017).
5. F. Le Louër and M.L. Rapún, Topological derivatives for solving inverse multiple scattering problems in threedimensional electromagnetism. Part II: Iterative method, SIAM J. Imaging Sci., 11, 734-769 (2018).
6. A. Carpio, T.G. Dimiduk, F. Le Louër and M.L Rapún. When topological derivatives met regularized Gauss-Newton
iterations in holographic 3D imaging, J. Comput. Phys., 388, 224-251 (2019).
7. M. Higuera, J.M. Perales, M.L. Rapún and J.M. Vega. Solving inverse geometry heat conduction problems by
postprocessing steady thermograms, Int. J. Heat and Mass Transfer, 143, art. 118490 (2019).

13

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

A SIMPLE EXAMPLE OF HÖLDER-LOGARITHMIC STABILITYIN INVERSE PROBLEMS
Michail Isaeva and Roman G. NovikovbaSchool of Mathematics, Monash University, 3800 Victoria, Australia, mikhail.isaev@monash.edubCMAP, Ecole Polytechnique, 91128 Palaiseau, France, roman.novikov@polytechnique.edu

Abstract
Many important inverse problems are exponentially ill-posed in general, which constitutes a severe difficulty fornumerical treatments. However, a stable reconstruction of the unknown parameter might still be possible in some cases
when the parameter is well-behaved. Countless results in the literature confirm improved stability under various
additional a-priori information. In fact, the behaviour of the stability bounds changes dramatically from the logarithmic
type to the Hölder type or even, under some strong assumptions, to the Lipchitz type. In this talk, we illustrate suchtransitions with an example from the classical Fourier analysis.

14

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

THE IDENTIFICATION OF A TIME-DEPENDENT SOURCEIN A TIME-FRACTIONAL DIFFUSION EQUATION WITH NON-SMOOTH SOLUTIONS
Karel Van BockstalaaDepartment of Electronics and Information systems, Ghent University, Krijgslaan 281 - S8,
Ghent, Belgium, karel.vanbockstal@ugent.be

Abstract
In this talk, an inverse source problem (ISP) for a time-fractional diffusion equation of order $\alpha\in(0,1)$
where the coefficients of the elliptic operator are dependent on spatial and time variables is discussed. The missing
solely time-dependent source is recovered from an additional integral measurement. First, the uniqueness of a solution
to the ISP will be shown. Next, two numerical algorithms will be proposed based on Rothe's method over uniform andgraded grids, and the convergence of iterates towards the exact solution will be discussed. An essential feature of thefractional subdiffusion problem is that the solution lacks smoothness near the initial time, although it would be smoothaway from $t = 0$. Rothe's method on a uniform grid addresses the existence of such a solution (non-smooth with$t^\gamma$ term where $1>\gamma > \alpha$) under low regularity assumptions, whilst Rothe's method over gradedgrids has the advantage to cope better with the behaviour at $t=0$ (also here $t^\alpha$ is included in the class ofadmissible solutions) for the considered problems. Numerical experiments will support these theoretically obtained
results.
This talk is based on joint work with A.S. Hendy (Ural Federal University).

15

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

PROPERTIES OF THE STEKLOV ZETA FUNCTION OF A SMOOTH PLANAR DOMAIN
OVER THE REAL AXIS
Alexandre Jollivet
Laboratoire de Mathématiques Paul Painlevé, CNRS UMR 8524/Université Lille 1 Sciences et Technologies, 59655
Villeneuve d'Ascq Cédex, France, alexandre.jollivet@math.univ-lille1.fr

Abstract
We address the question of reconstructing a bounded smooth and simply connected planar domain Ω from thespectrum of its Dirichlet-to-Neumann operator (Steklov spectrum). We state analog formulations of this problem, oneof them arising from Electrical Impedance Tomography.
First we review estimates from below for the Steklov spectral zeta function   on the real intervals(,1) and (1, ) Additional estimates are then given for the values of   at the negative even integers,
the so-called zeta invariants introduced in [1]. A compactness theorem for isospectral families of planar domains isderived, which generalizes a previous pre-compactness result given in [2].

Then we introduce a first variation formula for   when one smoothly deformes the domain  . We extends

previous estimates for   to the interval (1,1) by choosing an appropriate path of deformation. These results (0) .
also generalize a result in [3] on the derivative  
We also discuss application of the first variation formula to the question of nonexistence of nontrivial isospectral
deformations of a planar domain.
This talk is based on joint works [4,5,6] and works in progress with Vladimir Sharafutdinov (Novosibirsk State
University & Sobolev Institute of Mathematics).

References
1. E. Mal'kovich and V. Sharafutdinov, Zeta-invariants of the Steklov spectrum of a planar domain, Siberian Math. J.,
56(4), 678-698 (2015).
2. J. Edward, Pre-compactness of isospectral sets for the Neumann operator on planar domains, Commun. in PDE's,
18(7-8), 1249-1270 (1993).
3. J. Edward and S. Wu, Determinant of the Neumann operator on smooth Jordan curves, Proc. Amer. Math. Soc.
111(2), 357–363 (1991).
4. A. Jollivet and V. Sharafutdinov, On an inverse problem for the Steklov spectrum of a Riemannian surface. Inverse
problems and applications, Eds. P. Stefanov, A. Vasy, M. Zworski, Contemporary Mathematics 615, 165-191 (2014).
5. A. Jollivet and V. Sharafutdinov, An inequality for the zeta function of a planar domain, J. of Spectral Theory, J.
Spectr. Theory 8(1), 271–296 (2018).
6. A. Jollivet and V. Sharafutdinov, Steklov zeta-invariants and a compactness theorem for isospectral families ofplanar domains, J. Funct. Anal. 275(7), 1712–1755 (2018).

16

MINISYMPOSIUM
M2: New Trends in Regularization Theory
Organizers:
Bernd Hofmann, Technische Universität Chemnitz, Germany, bernd.hofmann@mathematik.tu-chemnitz.de
Stefan Kindermann, Johannes Kepler University Linz, Austria, kindermann@indmath.uni-linz.ac.at
Due to the ill-posedness of most linear and nonlinear inverse problems one needs regularization
techniques for their stable approximate solution. The theory is well developed for problems in Hilbert spaces
and during the last years many results have also been achieved for problems in Banach spaces. However,there are still many challenging questions, and permanently new classes of inverse problems occur,
motivated by applications from natural sciences, engineering and finance.
We want to bring together experts and young researches working in this field to discuss about new
results in the analysis and numerics of inverse and ill-posed problems.

17The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

A MODIFIED ASYMPTOTICAL REGULARIZATOIN OF NONLINEAR ILL-POSEDPROBLEMS
Christine Böckmann and Pornsarp Pornsawad
Potsdam University, Germany, bockmann@rz.uni-potsdam.de

Abstract
In this talk, we investigate the continuous version of modified iterative Runge–Kutta-type methods fornonlinear inverse ill-posed problems. The convergence analysis is proved under the tangential cone condition, amodified discrepancy principle, i.e., the stopping time T is a solution of F ( x  ( ))  y    for some

 *   and an appropriate source condition.
We yield the optimal rate of convergence.

18

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

ON THE DEAUTOCONVOLUTION PROBLEM IN THE TWO-DIMENSIONAL CASE
Yu Deng
TU Chemnitz, Research Group Regularization, Faculty for Mathematics, Chemnitz, 09107, Germany

Abstract
In the past years, there have been many contributions to the theory and practice of deautoconvolution as anonlinear inverse and ill-posed problem, with applications to stochastics, physics and other natural sciences. All thesecontributions have focused on the one-dimensional case, aimed at the stable approximate reconstruction of real orcomplex functions over some finite interval based on noisy observations of the autoconvolution of the desired function.
The complex case was discussed in [1] for an application in laser optics.
In the present talk, we try to present first ideas and results of numerical case studies concerning the twodimensional case. This case occurs when images (e.g., pictures over the unit square) have to be reconstructed fromcorresponding two-dimensional autoconvolution functions.

Reference[1] S.W. Anzengruber, S. Bürger, B. Hofmann, G. Steinmeyer, Variational regularization of complex
deautoconvolution and phase retrieval in ultrashort laser pulse characterization. Inverse Problems, 32 (3), 035002(27pp), 2016.

19

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

A NEW WAY OF INTERPRETING TIKHONOV REGULARIZATION AND ITSCONSEQUENCE FOR THE ESTIMATION OF SOLUTION SMOOTHNESS AND NOISE
LEVEL
Daniel GerthaaTechnical University Chemnitz, Faculty for Mathematics,Chemnitz, 09107, Germany

Abstract
In the theory for the regularization of ill-posed inverse problems, two parameters are essential for thedescription of regularization properties: noise level and a source condition, the latter characterizing the interplay
between solution smoothness and smoothing of the forward operator. Unfortunately, in many practical problems neitherof the parameters is available. In this talk we present an approach that allows to estimate the solution smoothness interms of an exponent in a source condition and the noise level. The method is based on how Tikhonov regularization
approximates an unknown solution. We show that one can use the established concept of approximate source
conditions and generalize them to not only describe the approximation of the unknown, but also the perturbed data andthe noise component. Using the relation between the approximation functions, we derive in a new way the well-known
results on optimal convergence rates and parameter choice rules but also reveal new relations between them.
The new approach is not restricted to Tikhonov regularization, we demonstrate that Landweber iteration
follows the same general principles.

20

The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS­2022)
Malta, May 22 ­ 28, 2022

ON REGULARIZED PROJECTION METHODS FOR ILL­POSED PROBLEMS
U. Hämarik, U. Kangro
Institute of Mathematics and Statistics, University of Tartu, Estonia
uno.hamarik@ut.ee, urve.kangro@ut.ee

Abstract
We consider in Hilbert spaces H, F linear ill­posed problem
A ∈ L(H, F ),Au = f,f ∈ R(A) ̸= R(A).

(1)

Only noisy data f δ satisfying ∥f − f δ ∥ ≤ δ are available . Let Pn , Qn (n ∈ N) be the orthoprojectors in
H, F respectively, satisfying ∥A(I − Pn )∥ → 0, ∥(I − Qn )A∥ → 0 (n → ∞). The projection methodfor (1) has the form An un = Qn f δ , An = Qn APn , un ∈ R(Pn ). We regularize this equation using
generating function gr : [0, a] → R (∥A∥ ≤ a, r ≥ 0) with properties|gr (λ)| ≤ γr
λp |1 − λgr (λ)| ≤ γp r−p(0 ≤ λ ≤ a,(0 ≤ λ ≤ a,

γ = const),

0 ≤ p ≤ p0 ,p0 > 1/2,

γp = const).

Let u0 ∈ H and let u∗ be the solution of (1), nearest to u0 . We findun,r = (I − gr (A∗n An )A∗n An )Pn u0 + gr (A∗n An )A∗n f δ .
For the choice of r we choose b2 ≥ b1 ≥ 1 and use the following discrepancy principle. If ∥An u0 −Qn f δ ∥ ≤ b1 δ, choose r = 0. Otherwise define r := supq>0 [(2q)1/q ∥(I − Pn )|A|q ∥−2/q ] with |A| =(A∗ A)1/2 and choose r ∈ (0, r) such that b1 δ ≤ ∥An un,r −Qn f δ ∥ ≤ b2 δ. If the last condition is violated
for all values r ≤ r, choose r = r or r = ⌊r⌋ + 1.
This choice of r guarantees convergence un,r → u∗ (δ → 0, n → ∞) and under assumptionsu∗ = |A|p z,∥z∥ ≤ ρ,u∗ − u0 = |A|p v,∥v∥ ≤ ρ,p ≤ 2p0 − 1(2)also the error estimate
1p∥un,r − u∗ ∥ ≤ const[ρ p+1 δ p+1 + ρ∥(I − Pn )|A|p ∥ + ρeµ (Qn )],{whereeµ (Qn ) =p∥(I − Qn )|A∗ |µ ∥min( µ ,2) ∀µ ∈ (0, 1],∥(I − Qh )A∥∥(I − Qh )|A∗ |p−1 ∥(3)µ ̸= p/2 if p ≤ 2;
if p ≥ 2.

Consider regularization of the projected equation by the m times iterated Tikhonov method with cor­1
responding generating function gr,m (λ) = [1 − ( 1+rλ)m ]/λ, p0 = m, m ≥ 1. Then for the choice of r
instead of the discrepancy principle the monotone error rule (ME­rule) can be recommended, replacing in
the rule formulated above the discrepancy ∥An un,r,m − Qn f δ ∥ by the ME­function(An un,r,m − Qn f δ , An un,r,m+1 − Qn f δ ).
∥An un,r,m+1 − Qn f δ ∥Then the error estimate (3) is true for p ≤ 2p0 = 2m (instead of the original p ≤ 2p0 − 1 = 2m − 1).
Note that obtained estimates for the discretization error are sharper than in well­known works (cf. [1]).
References
1. R. Plato and G. Vainikko, On the regularization of projection methods for soving ill­posed problems,
Numer. Math., 57, 63–79 (1990).

21

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

STOCHASTIC GRADIENT DESCENT IN BANACH SPACES
Zeliko Keretaa and Bangti Jinaa
University College London, Computer Science Department, Gower Street, London WC1E 6BT, UK,z.kereta@ucl.ac.uk;b.jin@ucl.ac.uk

Abstract
Stochastic gradient descent and its many variants are among the most promising and successful approaches forcomputing iterative solutions to optimization problems. They have outperformed traditional methods in many problem
settings and have seen increasing use for inverse problems. In each iteration, stochastic gradient methods use only asmall subset of data points to compute the estimator of the gradient. This results in methods that scale well to problem
size, which made them a popular choice for an increasing number of problems, and in particular they hold significant
promise for solving large-scale inverse problems.
Stochastic gradient methods are still predominantly used only in inner product spaces which provide a range of
geometric and analysis tools that facilitate their study. Convergence of SGD for solving linear inverse problems is inthese settings well understood [2]. However, the native domain of many inverse problems is in Banach spaces, such asparameter identification in elliptic PDEs, and Banach space norms are advantageous for e.g. preservation of sparsity.
Gradient descent-based reconstruction methods for solving linear inverse problems in Banach spaces in thedeterministic setting have been previously studied [1], and they typically follow Landweber type iterations
Mappings
are the so-called duality mappings, with parametersandsuch that. The search directionis the sub-gradient of the of thepower ofthe Banach space norm, lending a gradient descent interpretation of the iterates. Due to the duality mappings theiterates are non-linear. Moreover, instead of the Banach space norm it is more common to use the Bregman distance forthe convergence analysis. Due to these features of the problem, the analysis of the corresponding stochastic iterations
requires a set of tools and ideas different from Hilbert space settings, and different from the deterministic setting.
In this work we will present a mathematical framework for stochastic gradient descent in Banach spaces, and thecorresponding convergence analysis, for solving linear inverse problems. This is achieved by combining insights from
Hilbert space theory with novel approaches from modern stochastic tools for non-linear stochastic optimisation. The
theoretical findings shed further insights into the performance of the algorithm and open doors for a wide range of new
methods that we aim to study in future work.

References[1] F. Schopfer, A. K. Louis and T. Schuster, Nonlinear iterative methods for linear ill-posed problems in Banach
spaces, Inverse Problems., 22, 311-329 (2006).
[2] B. Jin and Xiliang Lu. On the regularising property of stochastic gradient descent. Inverse Problems, 35, 015004(2019).

22

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

LEARNED LANDWEBER ITERATION FOR THE TERAHERTZ TOMOGRAPHY

Clemens Meiser
University of Saarland, Germany, e-mail: meiser@math.uni-sb.de

Abstract
When talking about inverse problems deep learning and neural network based algorithms are an emerging
field. We consider the inverse problem of terahertz tomography which is an imaging
technique for monitoring plastics and ceramics. Starting with the nonlinear eikonal equation as physical model wetackle the problem by a learned Landweber iteration. The eikonal equation can be seen as a high frequency
approximation of the Helmholtz equation and, more generally, of the wave equation. Using the Landweber iteration we
have to solve the partial differential equation for the forward operator in every step, but also need to compute theadjoint operator of the Fréchet derivative. We reduce the computing time by learning both, the forward operator as wellas the adjoint of the Fréchet derivative.
Consequently, first we investigate the eikonal equation with respect to the theory of inverse problems. Then we
integrate neural networks in the Landweber iteration. In the talk we try to combine the theory of inverse problems withthe practical use of neural networks.
This is joint work with Marcel Mayr (SKZ Würzburg), Prof. Dr. Thomas Schuster and Dr. Anne Wald.

23

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

CONVERGENCE RESULTS FOR NONLINEAR TIKHONOV REGULARIZATION WITHOVERSMOOTHING PENALTY
Robert Plato
Department of Mathematics, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany,robert.plato@uni-siegen.de

Abstract
For solving nonlinear problems in Hilbert scales, we present results on Tikhonov regularization withoversmoothing penalty term, i.e., the exact solution of the nonlinear problem does not belong to the domain of
definition of the considered penalty functional. The focus of the present work is to provide convergence results for thissetting, without assuming any smoothness of the solution of the considered nonlinear problem. This work continues
previous studies, where convergence rates are provided for a priori- and a posteriori parameter choice strategies,
respectively, under certain smoothness assumptions on the solution.
This is joint work with B. Hofmann (TU Chemnitz).

References
1. B. Hofmann and R. Plato, Convergence results and low-order rates for nonlinear Tikhonov regularization withoversmoothing penalty term. ETNA 53, 313–328 (2020)24

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

GENERAL HEURISTIC RULE FOR CHOOSING REGULARIZATION PARAMETER INTIKHONOV METHOD
Toomas Raus
Institute of Mathematics and Statistics, University of Tartu, Narva mnt 18, Tartu, 51009, Estonia

Abstract
We consider an operator equation

Au  f *where A  LH , F  is the linear continuous operator between real Hilbert spaces H and F . We assume that insteadof the exact right-hand side

f * we have only an approximation f  F . To get regularized solution we consider

Tikhonov method u  I  A A*

 A f . Well-known heuristic rule for choosing regularization parameter   0
1*is the quasioptimality principle, where the parameter is chosen as the global minimum point of the function

 Q     du / d

on

the

set

of

parameters   j :  j  q j 1 , j  1,2,...,M , 0  q  1 .

Unfortunately, this rule is unstable in this sense that it often fails in case of heat-type problems. To get stable parameterchoice rules we introduce modified quasioptimality criterion function in the form  MQ  0  :  Q  0  , MQ  j  : max  Q  j , d MD  j  d MD  j 1  3 MQ  j 1  , j  1,2,...,M ,2where the function d MD    B  Au  f  , B  

1/ 2

aI  AA 

* 1 / 2.

General heuristic rule. For the regularization parameter choose the parameter H  max  low , min  MQ , upp .







For the parameter  MQ we can take the global minimum point of the function  MQ   on the set  or on the set

Lmin , where Lmin is the set of the local minimum points of the function  Q   . Another possibility is to find theparameter  MQ using triangle area rule [1] with the function  MQ   .

For the parameter  low we can take the global minimum point of the function  Q   or the parameter, which is foundby triangle area rule with the function  Q   . For the parameter  upp we can take global minimum point of the    1/ 2 d MD   or  DR     1/ 2 Au  f .

functions  HR 

All rules defined in such way are stable and they differ from each other only in the accuracy of the rule and thecomplexity of algorithm.
This work was supported by Estonian Research Council team grant PRG864.

References
1. T. Raus and U. Hämarik, Q-curve and area rules for choosing heuristic parameter in Tikhonov regularization. Mathematics
8 (7, 1166), 1–21 (2020).

25

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

CONVERGENCE ANALYSIS OF (STATISTICAL) INVERSE PROBLEMS UNDERCONDITIONAL STABILITY ESTIMATESFrank Werner
Julius-Maximilians-Universitat Wurzburg, Germany, frank.werner@mpibpc.mpg.de

Abstract
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if
the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse
problems in Hilbert scales satisfying conditional stability estimates characterized by general concave index functions.
For that case, we exploit Tikhonov regularization and provide convergence and convergence rates of regularized
solutions for both deterministic and stochastic noise.
We further discuss apriori and a posteriori parameter choice rules and illustrate the validity of our assumptions
in different model and real world situations.

26

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

A MIXED REGULARIZATION METHOD FOR ILL-POSED PROBLEMS
Wensheng Zhanga,baLSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
China, zws@lsec.cc.ac.cnbSchool of Mathematical Sciences, University of Chinese Academy Sciences, Beijing 100049, China

Abstract
We propose a mixed regularization method for ill-posed problems. The method combines iterative
regularization methods and parameter regularization methods effectively. First it applies iterative regularization
methods in which there is no continuous regularization parameter to solve the normal equation of the ill-posed problem.
Then continuous parameter regularization methods are applied to solve its residual problem. We consider the following
system(1)
T ( x)  y  ,where T : X  Y is a bounded linear operator, X and Y are two Hilbert spaces, y

is the known data with a noise level  . After (1) has been solved by an iterative regularization method yielding thesolution x , there is still a residual term y  Tx . Then we apply a parameter regularization method to solve
the residual equation. The final solution is the summation of the solutions of the iterative regularization method and theparameter regularization method and can be expressed as(2)
x mix  x it  g (T *T )T * ( y  y  ) ,
it

where

it

x m ix is the solution of the mixed regularization method, g is at least piecewise continuous function

associated with  and its detail description can be found in [1]. When the noise level  is
known, we can choose the regularization parameter by a posterior method such as ME-rule [2]. The ME-rule chooses
the smallest regularization parameter    ME for which we can guarantee the ME-property: the error of thesolution in this stage is monotonically increasing for   ME ,   . We have proved that the
it

m ix

solution x
obtained by the mixed regularization method is better than x by the iterative regularization method.
Our method is a combination of iterative regularization methods and parameter regularization methods. The
iterative regularization methods are such methods that they do not need continuous regularization parameters whereas
parameter regularization methods need. In iterative methods such as the Landweber method, the stopping rule is crucial
since the phenomenon of semi-convergence. The mixed regularization method can be considered as an improvement of
the continuous parameter regularization method. In continuous parameter regularization methods, the choice of theregularization parameter is crucial because the solution depends on the parameter sensitively. Our theoretical analysis
and many numerical examples show that the new mixed regularization method can reduce the sensitivity of theregularization parameter and improve the solution of iterative regularization methods. It can reach the optimal
convergence order under a much wider range.
This is a joint work with Hui Zheng.

References
1. H.W. Engl, M. Hanke, A.Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 2000.
2. U. Hamarik, U.Kangro, R. Palm, T.Raus, and U. Tautenhahn, Monotonicity of error of regularized solution and its
use for parameter choice, Inverse Problems in Science and Engineering, 22, 10-30 (2014).

27

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

SPLITTING FOR JUMP-DIFFUSION CALIBRATION IN FINANCIAL OPTION MODELS
Jorge P. Zubellia,ba
IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil, zubelli@gmail.com;b
Department of Mathematics, Khalifa University, P.O. Box 12788 , Abu Dhabi, AD, UAE, jorge.zubelli@ku.ac.ae

Abstract
Dupire’s local volatility model is extensively used and well-recognized for hedging and option pricing in financial
markets [1]. The inverse problem consists in recovering the time and space varying diffusion coefficient in a parabolic
partial differential equation from limited data. It is known that this corresponds to an ill-posed problem. Such ill-posedcharacter of local volatility surface calibration from market prices requires the use of regularization techniques either
implicitly or explicitly. Such regularization techniques have been widely studied for a while and are still a topic ofintense research [2-5]. We have employed convex regularization tools and recent inverse problem advances to deal
with the local volatility calibration problem. See [2-6] and references therein.
This talk concerns the calibration of Dupire's model in the presence of jumps. This leads to an integro-differential
equation whose parameters have to be calibrated so as to fit market data. We present a splitting strategy to identify
simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying
stochastic model consists of a jump-diffusion driven asset with time and price dependent volatility.
Our approach uses a forward Dupire-type partial-integro-differential equation for the option prices to produce aparameter-to-solution map. The ill-posed inverse problem for such a map is then solved by means of a Tikhonov-type
convex regularization. We present numerical examples that substantiate the robustness of the method both for synthetic
and real data.
This talk is partially based on joint work with Vinicius Albani (UFSC, Brazil) published in [6]. The research was
supported by FAPERJ under grant E26/202.927 and by Khalifa University under FSU-2020-09.

References
1. B. Dupire, Pricing with a smile, Risk, 7 (1994), 18–20.
2. A. De Cezaro, O. Scherzer and J. P. Zubelli, Convex regularization of local volatility models from option prices:convergence analysis and rates, Nonlinear Analysis, 75 (2012), 2398–2415.
3. V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math.,
8 (2014), 243–268.
4. A. De Cezaro and J. P. Zubelli, The tangential cone condition for the iterative calibration of local volatility surfaces,
IMA Journal of Applied Mathematics, 80 (2015), 212–232.
5. Y. F. Saporito, X. Yang, J. P. Zubelli The calibration of stochastic local-volatility models: An inverse problem
perspective, Computers & Mathematics with Applications 77:12 (2019), 3054-3067
6. V.V.L Albani, J. P. Zubelli, A splitting strategy for the calibration of jump-diffusion models, Finance andStochastics, 24:3 (2021), 677-722.

28

MINISYMPOSIUM
M3: Tomographic Inverse Problems
Organizers:
Anuj Abhishek, University of North Carolina, Charlotte, USA, anuj.abhishek@uncc.edu
Jan Boman, Stockholm University, Sweden, jabo@math.su.se
Venkateswaran P. Krishnan, Tata Institute for Fundamental Research, India, venkyp.krishnan@gmail.com
Eric Todd Quinto, Tufts University, USA, Todd.Quinto@tufts.edu
Tomography has revolutionized diagnostic medicine, nondestructive evaluation and materials
science. By nature, tomography is an inverse problem—recovering information about the structure orproperties of an object using indirect data. Modern modalities include novel methods in X-ray CT, hybrid
imaging, multi-modal imaging, multi-energy CT, ultrasound, Compton CT, and time-dependent problems.
Each modality generates novel mathematics and new algorithms, many of which will be presented in thisminisymposium.
We will bring together young and established researchers from the inverse problems community topresent their recent work and to foster discussion among participants.

29

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

INVERTIBLE DISTRIBUTIONS, MEAN VALUE OPERATORS AND SYMMETRICSPACES
Fulton Gonzalez
Tufts University, fulton.gonzalez@tufts.edu

Abstract
Integral transforms that are translation-invariant can be thought of as convolution operators. Therefore, to study
the mapping properties of such integral transforms, it is often useful to study convolution operators in general. One
such interesting type of translation-invariant integral transform is the fixed-radius mean value operator Mr on Rn. Moregenerally, one can study orbital integrals on homogeneous manifolds as either convolution operators or group-invariant
Radon transforms.
In the late 1950's and early 1960's Leon Ehrenpreis introduced the idea of slowly decreasing functions to describe
the Fourier transforms of compactly supported distributions S on Rn for which the convolution operator
has afundamental solution and is surjective on various function and distribution spaces. We will discuss this criterion indetail and some of its applications to mapping properties of mean value operators, as well as some recent related results
on wave equations as well as extensions to compact and noncompact symmetric spaces.

30

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

ESTIMATING HYPERPARAMETERS IN HIERARCHICAL BAYESIAN LINEAR
INVERSE PROBLEMS
K. Hall-Hoopera, A. Saibabaa, J. Chunglb and S. MillercaDepartment of Mathematics, North Carolina State University, Raleigh, NC 27695, USA,kahallho@ncsu.edu;aDepartment of Mathematics and Computational Modelling, Virginia Tech, Blacksburg, VA 24061, USA,jmchung@vt.edu;aDepartment of Environmental Health and Engineering, Johns Hopkins University, Baltimore, MD
21205, USA, smill191@jhu.edu

Abstract
We consider a hierarchical formulation of the Bayesian inverse problems in which the forward operator isassumed to be linear, and we treat unknown parameters associated with the noise and the prior as hyperparameters. We
marginalize over the unknown parameters and compute the maximum a posteriori estimator over the marginalized
distribution. We use Krylov subspace methods to accelerate the computations of the nonlinear optimization technique.
We demonstrate the performance of our approach on several synthetic test problems.

31The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

ALGORITHMS FOR THE COMPTON SCATTER IMAGING IN POSITRON EMISSIONTOMOGRAPHY
Ivan G. Kazantsev
Institute of Computational Mathematics and Mathematical Geophysics, 6 Lavrentiev street, 630090 Novosibirsk,
Russia, kazantsev.ivan6@gmail.com

Abstract
We investigate a forward model of the Compton scatter in PET using the SSS technique [1] that estimates a single
scatter in the detector pair ( A, B) as an integral over the total scatter volume V :
AB
V

S

S

B

S

B

A

S

A

S

 (   ' dl    dl ) B  C A B A  (   dl    ' dl ) S  dV( I  I ); I  e
fdl ; I B  e22S fdl.
4 | AS | | SB |  C VA

 AS SB

Here,

(1)

 AS and  BS are the geometric cross-sections of the detectors A and B , f is the emitter activity,   ( E, S ) is the linear attenuation coefficient depending on the energy E and the scatter point S , and  C / is the differential cross-section. Primed and unprimed quantities are evaluated at the scattered and unscattered photon
energies, respectively. Equation (1) is symmetrical in terms of A and B so that the primary photons are recorded bothby A and B . We investigate a one-sided version of (1), where only the detector A is tuned to counting the primary
photons, that is, we set I  0 within idealized assumptions of the excellent energy resolution and detection
efficiency for the detectors A and B . Then (1) can be reduced [2] (in spherical coordinates with origin in A ) to:
B



AB

S

B

cos  cos(   ) ( ,  ,| AS |)  C  ( A  dl  S  ' dl )  dde00 f ( , , r )dr.
4 | AB |
C

0


2

| AS |(2)

The kernel support has a spindle shape, and the kernel larger values are concentrated around the detector A ,rapidly decreasing in the direction to the detector B . It also rapidly decreases in the lateral direction, from the vicinityof the central line AB to the kernel borders, so that the kernel gives larger weights to the activity for the areas close to
A and smaller weights to the kernel domain around B . There emerges additional information from a scatter about theactivity hot spots located close to the patient’s body periphery. We reduce (2) to the slice-by-slice convolution distancedependent blurring model of the projection formation provided that the attenuation map is assumed to be constant. Thismodel has been proven earlier as invertible [3]. We show that the three-dimensional slice-by-slice filteredbackprojection algorithm developed in [3] is applicable to the scatter data inversion. Because of the idealized
assumptions involved in the derivation of the analytical scatter projector (2), we validate this model with Monte Carlosimulation. The numerical comparative study has been performed using a digital cylindrical phantom filled with watercontaining spherical activity sources. We present the results illustrating the numerical experiments conducted.

References
1. C. C. Watson, D. Newport and M. E. Casey, A Single Scatter Simulation Technique for Scatter Correction in 3D
PET, in: Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Kluwer, Dordrecht, 255-268(1996).
2. I. G. Kazantsev, U. L. Olsen, H. F. Poulsen and P. C. Hansen, A spectral geometric model for Compton single scatterin PET based on the single scatter simulation approximation, Inverse Problems, 34, 024002 (2018).
3. I. G. Kazantsev, J. Klukowska, G. T. Herman and L. Cernetic, Fully three-dimensional defocus-gradient corrected
backprojection in cryoelectron microscopy, Ultramicroscopy, 110, 1128-1142 (2010).

32

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

SPARSITY-BASED TECHNIQUES FOR HYBRID IMAGING MODALITIES WITHMISSING LOW FREQUENCIES

Leonid Kunyansky
University of Arizona, USA, leonk@math.arizona.edu

Abstract
Several novel hybrid imaging modalities rely on propagating sound waves to deliver spatial information about
the properties of the object. Traditional image reconstruction techniques assume that an infinitely wide range of
acoustic wavelengths is available. However, commonly utilized piezoelectric transducers emit/receive only a narrow
band of frequencies, with both very long and very short wavelengths missing. This leads to strong distortions in thereconstructed images.
We will investigate the use of sparsity‐inducing techniques (such as, for example, L1‐regularization) to
overcome this problem.
This is a joint work with N. Do and P. Hoskins.

33

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

NOVEL INVERSE PROBLEMS IN COMPTON TOMOGRAPHY

Eric Todd Quinto
Tufts University, todd.quinto@tufts.edu

Abstract
In this talk, we will describe novel luggage testing methods using Compton CT. We will analyze their microlocal
properties and describe possible added artifacts and visible and invisible singularities of the objects.
We will outline reconstruction methods that suppress the artifacts and provide reconstructions illustrating our
results.

34

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

NUMERICAL RECONSTRUCTION FROM THE FOURIER TRANSFORM ON THE BALL
VIA PROLATE SPHEROIDAL WAVE FUNCTIONS

G. Sabinin
Faculty of Mechanics and Mathematics, Lomonosov MSU, 1 Leninskiye Gory, 119991 Moscow, Russia,
gvsabinin@gmail.com

Abstract
We implement numerically formulas of [1] for finding a compactly supported function v on Rd, d ≥ 1, from its
Fourier transform Fv given within the ball Br.
For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, whicharise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1.
In multidimensional case, these formulas also include inversion of the classical Radon transform.
This talk is based on the joint work with M. Isaev and R.G. Novikov.

References
1. M. Isaev and R.G. Novikov, Reconstruction from the Fourier transform on the ball via prolate spheroidal wave
functions. 2021. ffhal-03289374

35

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

RECONSTRUCTION OF SPARSE LOG-CONDUCTIVITY IN CURRENT DENSITYIMPEDANCE IMAGING

Souvik Roy
University of Texas at Arlington, USA, souvik.roy@uta.edu

Abstract
In this talk, we present a new approach to reconstruct sparse log-conductivities in an object through current
density impedance tomography (CDII). CDII is a non-invasive hybrid imaging modality where voltage is passed toelectrodes placed across an object and the corresponding interior electric field is measured. Using these measurements,
one tries to reconstruct the interior of the object. Unfortunately, existing mathematical reconstruction algorithms sufferfrom loss of contrast in the images. We, thus, use an alternate method comprising of sparse optimization and edgeenhancing anisotropic diffusion to obtain images with high quality.
Several numerical experiments demonstrate the superiority

36

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

SYMMETRY FROM SECTIONAL INTEGRALS FOR CONVEX DOMAINS

Suman Kumar Sahoo
TIFR Centre for Applicable Mathematics, India, suman@tifrbng.res.in
Department of Mathematics. and Statistics, University of Jyvaskyla, Finland, suman.k.sahoo@jyu.fi

AbstractLet  be a bounded convex domain in R ( n  2 ). In this work, we prove that if there exists an integrable
function f such that it's Radon transform over (n  1) -dimensional hyperplanes intersecting the domain  is anfunction G depending on the distance to the nearest parallel supporting hyperplane to  , then  is a ball andfis radial depending on certain assumptions on G . As a consequence we show that constants are not in range of Radontransform of integrable functions in dimensions n  3 .
This work jointly done with a fellow PhD student Mr. Ramya Dutta.

37

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 -28, 2022

FUNCTIONAL-ANALYTICAL METHODS IN ACOUSTIC INVERSE ROBLEMS
Andrei Shurup
Lomonosov Moscow State University, Faculty of Physics, Acoustics Department, Moscow, Russia
Sсhmidt Institute of Physics of the Earth of the Russian Academy of Sciences, Moscow, Russia
e-mail: shurup@physics.msu.ru

Abstract
In this report possibilities of functional algorithms [1–3] for purposes of 2D and 3D acoustic tomography are
discussed. Initially, these algorithms have been developed in quantum mechanics. To understand applicability of thesemethods for acoustics inverse problems the detail investigations based on numerical modeling were implemented [4-6].
For acoustic applications it is important that these algorithms take into account the multiple scattering processes and do
not require iterations or additional regularizations. Moreover, the joint reconstruction of both scalar (i.e. sound speed,
absorption coefficient, density) and vector (i.e. flows) characteristics of the medium is possible in a single tomography
scheme, without additional separation of influence of inhomogeneity components on the scattering data.
In 2D case the tomography scheme for joint reconstruction of spatial distributions of sound speed, density,
absorption together with recovering the frequency dependence of the absorption is numerically studied. It is shown, thatthe interference resistance of this algorithm is quite high that indicates the possibility of its practical application for
medical purposes (for example, in the early diagnostics of cancer in mammology), taking into account the abilities ofreal tomographic devices.
Numerical results of 3D reconstruction are also presented for the purposes of ocean acoustic tomography. In thiscase 3D inverse problem is approximated by a set of 2D ones which take into account multi-channel scattering effects.
These results are perspective for the ocean tomography when nonadiabatic propagation of acoustic modes in ocean
waveguides should be taken into account. Results of numerical reconstructions of 3D ocean inhomogeneities based on
the 2D multi-channel functional-analyticla algorithm are presented, which show high resolution and acceptable forpractical applications the interference resistance of this algorithm.
The comparative numerical investigation of iterative algorithm [7] and the considered functional-analyticalalgorithm for solving 2D acoustic tomography problem is also considered. For the case of the two-dimensional scalarHelmholtz equation, the efficiency of the iterative algorithm for the reconstruction of medium-strength scatterers andthe advantages of the functional-analytical approach for the reconstruction of strong scatterers with parameters typicalfor acoustic tomography problems are demonstrated [8]. The main advantage of the iterative approach in comparison
with the functional-analytical algorithm is the ability to make reconstruction from incomplete scattering data. This isespecially important in 3D inverse problems.
The obtained results of numerical investigations of noise immunity and resolution show that the functionalanalytical algorithm can be applied in practical acoustic tomography problems, as well as the possibility of developing
new tomographic schemes based on iterative and functional-analytical approaches.
The reported study was funded by RFBR and CNRS, project number 20-51-15004.

References
1.
2.
3.
4.
5.
6.
7.
8.

R.G. Novikov, Proc. Steklov Inst. Math., 225 (1999), no. 2, 285–302.
R.G. Novikov, M. Santacesaria, Int. Math. Res. Notices, 6 (2013), 1205-1229.
A.D. Agaltsov, R.G. Novikov, J. Math. Phys., 55 (2014), no. 10, 103502.
V.A. Burov, N.V. Alekseenko, O.D Rumyantseva, Acoust. Phys., 55 (2009), no. 6, 843–856.
V.A. Burov, A.S. Shurup, O.D. Rumyantseva, Acoust. Phys., 59 (2013), no. 3, 345–360.
O.S. Krasulin, A.S. Shurup, Bull. Russ. Acad. Sci. Phys., 84 (2020), no. 2, 289-294.
R.G. Novikov, Sbornik: Mathematics, 206 (2015), no. 1, 120–134.
A.S. Shurup, Eurasıan Journ. of Math. and Comp. Appl., 10 (2022), no. 1.

38

MINISYMPOSIUM
M4: Modern Challenges in Inverse Problems Including Boundary Rigidity,
Microlocal Analysis and Cloaking
Minisymposium dedicated to the 70th anniversary of an outstanding expert in inverse
problems, Walker Family Endowed Professor of Mathematics University of Washington,
Gunther Uhlmann
Organizers:
Gen Nakamura, Hokkaido University, Japan, nakamuragenn@gmail.com
Eric Todd Quinto, Tufts University, USA, Todd.Quinto@tufts.edu
Plamen Stefanov, Purdue University, USA, stefanop@purdue.edu
Professor Gunther Uhlmann has made fundamental contributions for decades to microlocal analysis
and applications, the solution of important inverse problems including Electrical Impedance Tomography(EIT) also called Calderon's problem, travel time tomography, also called boundary rigidity and lens rigidity,
integral geometry, and inverse problems arising in general relativity. He and collaborators have done
pioneering work in proposing transformation optics, a method to make objects invisible to electromagnetic
waves, acoustic waves, matter waves, and other types of waves.
This minisymposium celebrates the anniversary of Professor Uhlmann, his many fundamental andinsightful contributions to inverse problems and partial differential equations, and his pioneering work in themathematics of boundary rigidity, microlocal analysis and cloaking. The methods pioneered by Prof.
Gunther Uhlmann are essential contributions to almost all areas of inverse problems.

39

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

THE SOLID-FLUID TRANSMISSION PROBLEM

Nikolas Eptaminitakis
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA,neptamin@purdue.edu

Abstract
I will discuss recent joint work with Plamen Stefanov ([1]) on the microlocal analysis of the transmission problem
at an interface between an isotropic linear elastic solid and an inviscid fluid. This problem is motivated fromgeophysics: there one is interested in understanding the propagation of seismic waves in the interior of the Earth, which
consists of several solid and fluid layers. When a seismic wave meets the interface between two layers, part of its
energy is reflected (possibly with mode conversion), and, if the angle of incidence is not too large, part of it istransmitted to the other side of the interface. In a spirit similar to previous works studying microlocally the transmission
problem at the interface between two solids ([2,3,4,5]), our goal is to understand reflection, transmission and mode
conversion of singularities at the interface between a solid and a fluid. For simplicity, we consider the case of twolayers, with the fluid layer being enclosed by the solid one. The two media are described by a system of PDEs modelingthe displacement in the solid and pressure-velocity in the fluid, with these quantities being coupled at the interface bytransmission conditions. We show well posedness for the system, and construct and justify a parametrix for it(approximate solution up to smooth error) using geometric optics.
As an application of our study, we consider the inverse problem of recovering the wave speeds in the two layers andthe material density in the solid outer layer from the Neumann-to-Dirichlet map for the solid-fluid systemcorresponding to the exterior boundary.

References
1. N. E. and Plamen Stefanov, The Solid-Fluid Transmission Problem, arXiv: 2111.03218
2. Sönke Hansen, Propagation of polarization in transmission problems, arXiv:2107.10200
3. Plamen Stefanov, Gunther Uhlmann, and András Vasy. The transmission problem in linear isotropic elasticity, Pure
Appl. Anal., 3(1):109–161, 2021.
4. Kazuhiro Yamamoto, Elastic waves in two solids as propagation of singularities phenomenon, Nagoya Math. J.,
116:25–42, 1989.
5. Kazuhiro Yamamoto, Reﬂective and refractive phenomena of tangential elastic waves to the boundary in two solids
as a propagation of singularities, J. Math. Pures Appl. (9), 92(2):188–208, 2009.

40

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

SOBOLEV ESTIMATES FOR MULTILINEAR RADON TRANSFORMS VIA PARTITION
OPTIMIZATION
Allan Greenleaf
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA

Abstract
I will describe a method for obtaining Sobolev space bounds for multilinear generalized Radon transforms andFourier integral operators by optimizing over collections of linear estimates. This approach was motivated by problems
in geometric measure theory/combinatorics, and I will give examples of these applications, but this technique might beapplicable for other problems involving multilinear operators, such as Born expansions.
This is joint work with Alex Iosevich and Krystal Taylor.

41

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

SIMULTANEOUS RECOVERY OF ATTENUATION AND SOURCE DENSITY IN SPECT

Sean Holman
The University of Manchester, United Kingdom, sean.holman@manchester.ac.uk

Abstract
I will discuss some recent results about simultaneous recovery of the attenuation and source density in the
SPECT inverse problem. Assuming the attenuation is piecewise constant and the source density piecewise smooth weshow that, provided certain conditions are satisfied, it is possible to uniquely determine both. I will also discuss anumerical algorithm that allows for determination of both parameters in the case when the range of the piecewise
constant attenuation is known.
This is based on joint work with Philip Richardson.

42

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INVERSE PROBLEMS FOR FINITE GRAPHS AND CLOAKING
Matti Lassasa , Emilia Blåstenb, Hiroshi Isozakic and Jinpeng Luaa
Department of Mathematics and Statistics, University of Helsinki, Finland;b
Department of Mathematics, Aalto University, Finland;c
Department Graduate School of Pure and Applied Mathematics, Tsukuba University, Japan

Abstract
We study the inverse problem of determining a finite weighted graph having vertexes X and edges Efrom the observations made on a finite subset of vertexes subset B  X . The observations we consider are1) the source-to-solution map on a vertex subset B  X for heat equation on a graph,2) eigenvalues and restrictions of eigenfunctions on B  X ,3) observations of a random walk moving on a graph.
We solve these problems when the graph satisfies the Two-Points Condition. This condition is valid for trees
and perturbations of finite lattices. We also discuss how the studied problems are related to Finite Element Method,resistor networks, and inverse problems on manifolds and invisibility cloaking.

References[1] Emilia Blåsten , Hiroshi Isozaki , Matti Lassas , Jinpeng Lu : Inverse problems for discrete heat equations andrandom walks, arXiv:2107.00494, 2021
[2] M. Lassas, M. Salo, L. Tzou: Inverse problems and invisibility cloaking for FEM models and resistor networks.
Mathematical Models and Methods in Applied Sciences 25 (2015), 309.
[3] A. Greenleaf, M. Lassas, G. Uhlmann: On nonuniqueness for Calderon&#39;s inverse problem, Mathematical
Research Letters 10 (2003), no. 5-6, 685-693

43

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INVERSE TRANSPORT AND DIFFUSION PROBLEMS IN PHOTOACOUSTIC IMAGING
WITH NONLINEAR ABSORPTION

Ru-Yu LaiaaSchool Of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA, rylai@umn.edu

Abstract
Two-photon photoacoustic tomography (TP-PAT) is a noninvasive optical molecular imaging modality. It
aims at inferring the two-photon absorption property of heterogeneous materials, such as biological tissues, fromphotoacoustic measurements. In this talk, we will discuss the work [1] motivated by TP-PAT on inverse coefficient
problems for a semilinear radiative transport equation and its diffusion approximation with internal data. These data arefunctionals of the coefficients and the solutions to the equations. Uniqueness and stability results of the inverse
problems as well as its uncertainty quantification will also be discussed.

References
1. R.-Y. Lai, K. Ren, and T. Zhou, Inverse transport and diffusion problems in photoacoustic imaging with nonlinear
absorption, SIAM Applied Mathematics, (2021).

44

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

WAVE PROPOGATION INSIDE TRANSPARENT SCATTERERS AND APPLICATIONS
Hongy Liu
City University of Hong Kong, Hong Kong, China

Abstract
Consider the time-harmonic wave scattering from an inhomogeneous medium scatterer. If
transparency/invisibility occurs, the incident and total wave fields form a pair of transmission eigenfunctions. We shalldiscuss our recent discovery on the spectral geometry of transmission eigenfunctions as well as its applications tosuper-resolution imaging, artificial mirage and pseudo polariton resonances.

References[1] Y.-T. Chow, Y. Deng, Y. He, H. Liu and X.Wang, Surface-localized transmission eigenstates,super-resolution
imaging and pseudo surface plasmon modes, SIAM Journal on Imaging Sciences, 14 (2021), no. 3, 946—975.
[2] Y. Deng, H. Liu, X. Wang and W. Wu, On geometrical properties of electromagnetic transmission eigenfunctions
and arti cial mirage, SIAM Journal on Applied Mathematics, 82 (2022), no. 1, 124.
[3] Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions,
arXiv:2109.14361[4] E. Blasten and H. Liu, On vanishing near corners of transmission eigenfunctions, Journal of Functional Analysis,
273 (2017), 3616-3632.

45

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INTERSECTION RIGIDITY FOR SIMPLE RIEMANNIAN MANIFOLDS

Reed Meyerson
University of Helsinki, Finland, reed.meyerson@helsinki.fi

Abstract
Consider a compact Riemannian manifold with boundary. For each pair of inward pointing unit vectors at theboundary, there is a corresponding pair of geodesics through the manifold. The “intersection data” is the collection of
all pairs of inward pointing directions whose geodesics have non-empty intersection. This data is equivalent toforgetting the travel time information from the broken scattering relation [1]. We show that the intersection data for a
simple Riemannian manifold determines its isometry class when the dimension of the manifold is at least 2.

References
1. Y. Kurylev, M. Lassas, and G. Uhlmann, Rigidity of Broken Geodesic Flow and Inverse Problems, American
Journal of Mathematics., 132, no. 2 (2010): 529-62

46

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

MAPPING PROPERTIES OF X-RAY TRANSFORMS NEAR CONVEX BOUNDARIESFrancois MonardaaDept. Of Mathematics, University of California Santa Cruz, Santa Cruz, CA, 95064, USA, fmonard@ucsc.edu

Abstract
On a Riemannian manifold with boundary, the X-ray transform integrates a function or a tensor field along all
geodesics through the manifold. The reconstruction of the integrand of interest from its X-ray transform is the basis ofimportant inverse problems with applications to seismology and medical imaging.
The inversion of the X-ray transform is often done by inverting the normal operator (composition of the Xray transform and its adjoint, the "backprojection" operator). This requires the design of appropriate Sobolev scales
or Frechet spaces over the manifold and the manifold of its geodesics, compatible with an appropriate formulationof forward and backward mapping properties for the X-ray transform, the backprojection operator, and theircomposites. For example, a landmark result by Pestov and Uhlmann, paving the way toward their 2005 result on theboundary rigidity of simple surfaces [1], was the design of a good Frechet setting where the backprojection operator issurjective.
In this talk, I will review recent results [2,3,4,5,6,7] attempting to shed additional light on the (forward andbackward) mapping properties of the X-ray transform and its normal operator(s) on convex, non-trapping manifolds.

References
1. L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid,
Annals of Mathematics, 161 (2005), pp. 1093–1110.
2. F. M., R. Nickl, and G. P. Paternain, Efficient Nonparametric Bayesian Inference for X-Ray Transforms, Annals of
Statistics 2019, Vol. 47, No. 2, 1113-1147.
3. F.M., Functional relations, sharp mapping properties and regularization of the X-ray transform on disks of constantcurvature, SIAM Journal of Math. Anal., 52(6), 5675—5702.
4. F. M., R. Nickl, and G. P. Paternain, Consistent inversion of noisy non-abelian x-ray transforms, Communications on
Pure and Applied Mathematics, 74 (2021), pp. 1045–1099.
5. F. M., R. Nickl, and G. P. Paternain, Statistical guarantees for bayesian uncertainty quantification in non-linear
inverse problems with gaussian process priors, The Annals of Statistics, 49 (2021), pp. 3255–3298.
6. R. Mazzeo and F. M., Double b-fibrations and desingularization of the x-ray transform on manifolds with strictly
convex boundary. arXiv preprint arXiv:2112.14904, 2022.
7. R. Mishra, F.M. and Y. Zou, The C-infinity property for a class of singularly weighted X-ray transforms. arXiv
preprint arXiv:2203.09861, 2022.

47

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

RECENT DEVELOPMENTS IN PHOTOACOUSTIC TOMOGRAPHY

Benjamin PalaciosaaDeptartment of Mathematics, Universidad Catolica de Chile, Santiago, Chile,
benjamin.palacios@mat.uc.cl

Abstract
Photoacoustic Tomography is a promising hybrid medical imaging modality that can generate high-resolution
and high-contrast images by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasoundwaves via the photoacoustic effect. One step of the mathematical problem consists in the determination of an acoustic
source from boundary observations. There exist several ways of addressing the reconstruction of the initial conditions,with varying performances depending highly on the underlying assumptions on the acoustic properties of the body, inparticular, on the hypotheses imposed on the speed of sound. Some examples include integral formulae and eigenvalue
expansion methods mainly for homogeneous backgrounds, while time-reversal based approximations and iterative
reconstruction procedures are better suited for acoustically heterogeneous media.
In the last 13 years, many new results have stemmed from the crucial work of Stefanov and Uhlmann between
2009 and 2013 [1,2,3], which provided a deep understanding of the underlying geometrical problem. Their microlocal
approach combined with functional analysis and geometry allowed them to answer the fundamental questions of
uniqueness and stability, and to obtain an iterative reconstruction formula in the form of a Neumann series. Since then,researchers have been able to extend and generalize to different settings their techniques and ideas, however, many
open questions still remain to be answered, especially in the more intricate problem of simultaneous determination of
the initial source and the sound speed with a single boundary measurement.
In this talk, I will review the main ideas introduced in [1,2,3] and I will talk about some of the latest
developments on this problem, in particular the results obtained in [4] related to the partial data case. I will also address
the aforementioned joint determination problem of recovering the initial source and the sound speed with a single
boundary measurement. We will review previous results as well as recent progresses made in collaboration with
Sebastian Acosta [5].

References
1. P. Stefanov and G. Uhlmann (2009), Thermoacoustic tomography with variable sound speed, Inverse Problems, 25
075011.
2. P. Stefanov and G. Uhlmann (2011), Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27045004.
3. P. Stefanov and G. Uhlmann (2013), Recovery of a source term or a speed with one measurement and applications.
Trans. Amer. Math. Soc. 365: 5737–5758.
4. B.P. (2022), Photoacoustic tomography in attenuating media with partial data. Inverse Problems and Imaging, doi:
10.3934/ipi.2022013
5. S. Acosta and B. P. (2022) Geometric conditions for the joint determination of source and wave speed inphotoacoustic imaging (working title), ongoing project.

48

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

FORMALLY DETERMINED INVERSE PROBLEMS FOR HYPERBOLIC PDES

RakeshDepartment of Mathematical Sciences, University of Delaware, Newark DE, USA
rakesh@udel.edu

Abstract
We describe two stability results for formally determined inverse problems for hyperbolic PDEs. Our first
result is Lipschitz stability for the fixed angle scattering problem for the operator $\Box + q(x)$, where the dataconsists of the medium response to incoming plane waves from two opposite directions.
This is joint work with Mikko Salo. Our second result is Lipschitz stability for the fixed angle scattering
problem for the operator $\Box + q(x)$, where the data consists of the medium response to plane waves coming fromthe same fixed direction but with different time delays.
This is joint work with Venky Krishnan and Soumen Senapati. For both these problems there are also results
for the operator $(\partial_t -a)^2 - (\nabla -b)^2 + c$, by Salo et al for the time independent case, and by Krishnan,
Rakesh and Senapati for the time independent case.
We also have similar results with point sources instead of plane wave sources.

49

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INSTABILITY MECHANISMS FOR INVERSE PROBLEMS

Mikko Salo
Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland, mikko.j.salo@jyu.fi

Abstract
Many inverse and imaging problems, such as image deblurring or electrical/optical tomography, are known to behighly sensitive to noise. Such problems are called ill-posed or unstable, as opposed to being well-posed (a notion
introduced by J. Hadamard in 1902). Instability lies at the heart of inverse problems research and much of the literatureaddresses various aspects of it.
The inherent reason for instability is easy to understand in linear inverse problems like image deblurring. For more
complicated nonlinear imaging problems the instability issue is more delicate. We will discuss a general framework forunderstanding instability in inverse problems based on smoothing/compression properties of the forward map together
with estimates for entropy and capacity numbers in relevant function spaces. The methods apply to various inverse
problems involving general geometries and low regularity coefficients.
This talk is based on joint work with Herbert Koch (Bonn) and Angkana Rüland (Heidelberg).

50

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

ON THE RANGE OF THE PLANAR X-RAY TRANSFORM OF SYMMETRIC TENSORSON THE FOURIER LATTICE OF THE TORUS

Kamran Sadika and Alexandru TamasanbaFaculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
kamran.sadiq@univie.ac.atb
Department t of Mathematics, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816, USA,tamasan @math.ucf.edu

Abstract
I will present new necessary and sufficient conditions on the Fourier coefficients of a function on a torus to be in
the range of the X-ray transforms of symmetric two tensors compactly supported in the unit disc.
The conditions can be construed to define a deficiency index for the X-ray data.

References
1. K. Sadiq and A. Tamasan, On the range of the planar X-ray transform on the Fourier lattice of the torus.
arXiv:2201.10926

51

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

ESTIMATE THE SIZE OF AN INCLUSION IN A BODY WITH COMPLEXCONDUCTIVITY USING FINITE NUMBER OF MEASUREMENTS

Jenn-Nan Wang
National Taiwan University, Taiwan, jnwang@math.ntu.edu.tw

Abstract
In the inverse boundary value problems of the second order elliptic equation with complex conductivity, wederive estimates for the volume fraction of an inclusion whose physical parameters satisfy suitable gap conditions.
Lower and upper bounds are obtained by using three pairs of boundary measurements. We accomplish this with thehelp of the "translation method" which consists of perturbing the minimum principle associated with the equation byeither a null-Lagrangian or a quasi-convex quadratic form.
This is a joint work with Catalin Carste.

52

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INTEGRAL GEOMETRY PROBLMES IN LORENTZIAN GEOMETRY

Yiran Wangaa
Deptartment of Mathematics, Emory University, Atlanta, GA, 30322, USA, yiran.wang@emory.edu

Abstract
We consider the light ray transform on Lorentzian manifolds, which concerns the integral of functions alonglight-like (or null) geodesics. The transform is related to the Radon transform or geodesic ray transform in the
Riemannian setting. An outstanding question is what information regarding the function can be recovered from the light
ray transform. In this talk, we discuss recent developments, including injectivity, stability results and microlocal
properties of the transform. Also, we discuss the applications of the transform to inverse problems in cosmology andLorentzian geometry. The talk is partly based on references [1-3].

References
1. A. Vasy, Y. Wang. On the light ray transform of wave equation solutions. Communications in Mathematical Physics
384.1 (2021): 503-532.
2. Y. Wang. Microlocal analysis of the light ray transform on globally hyperbolic Lorentzian manifolds. arXiv:
2104.08576 (2021).
3. Y. Wang. Some integral geometry problems for wave equations. arXiv:2109.14446 (2021).

53

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INVERSE PROBLEMS FOR NONLINEAR PDES

Ting Zhou
Zhejiang University, Hangzhou, China

Abstract
In this talk, I will demonstrate the higher order linearization approach to solve several inverse boundary value
problems for nonlinear PDEs modeling nonlinear electromagnetic optics including nonlinear time-harmonic Maxwell’sequations, nonlinear magnetic Schrodinger equation and its fractional version. The problem will be reduced to solving
for the coefficient functions from their integrals against multiple linear solutions. We will focus our discussion ondifferent choices of linear solutions.

References[1] R.-Y. Lai and T. Zhou, Inverse problems for nonlinear fractional magnetic Schrödinger equation, submitted,
arXiv:2103.08180.
[2] R.-Y. Lai and T. Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equation, accepted by
Math. Res. Lett., arXiv:2007.02475.

54

MINISYMPOSIUM
M5: Theoretical and Numerical Results in Geometric Inverse Problems for PDEs
Organizers:
Anna Doubova, University of Seville, Spain, doubova@us.es
Jone Apraiz, University of the Basque Country, Spain, jone.apraiz@ehu.eus
In recent years, the interest for the analysis and simulations of inverse problems for Partial
Differential Equations (PDE’s) of many kinds has grown a lot. This is motivated for their relevance inmany applications: elastography and medical imaging, seismology, potential theory, ion transport problems
or chromatography and finances, for example. The study of such type of problems includes theoreticalresults on the uniqueness, stability and numerical approximations.
The purpose of this minisymposium is to show and share different analytical and computational
methods for inverse problems that could be based in the problems that arise from Science, Medicine orTechnology.

55

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

GEOMETRIC INVERSE PROBLEMS FOR THE BURGERS EQUATION AND RELATEDSYSTEMS

Jone Apraiz
Mathematics Department, University of the Basque Country, Barrio Sarriena s/n, 48940 Leioa(Bizkaia), Spain, jone.apraiz@ehu.eus

Abstract
The analysis and solution of geometric inverse problems has recently increased a lot because of their importance inmany applications: elastography and medical imaging, seismology, potential theory, ion transport problems or
chromatography and other similar fields.
In this talk we will deal with one-dimensional inverse problems concerning the Burgers equation and some related
nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of thespatial interval from some appropriate boundary observations of the solution. Depending on the properties of the initialand boundary data, we will see uniqueness and non-uniqueness results. We will also show numerical solutions to theinverse problems mentioned above, computing accurate approximations of the interval sizes.
This is a joint research project in collaboration with Anna Doubova, Enrique Fernández-Cara and Masahiro
Yamamoto.

References
1. J. Apraiz, J. Cheng, A. Doubova, E. Fernández-Cara and M. Yamamoto, Uniqueness and numerical reconstruction
for inverse problems dealing with interval size search, Inverse Problems and Imaging, to appear (2021).
2. J. Apraiz, A. Doubova, E. Fernández-Cara and M. Yamamoto, Some Inverse problems for the Burgers equation and
Related Systems, to appear (2021).
3. M. Yamamoto, Lectures on inverse problems, Università Roma 2 “Tor Vergata" (2021).

56

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

INVERSE PROBLEMS FOR PARABOLIC EQUATION IN NETWORKS WITH LOOPS

Jone Apraiza and Jon Asier Bárcena-PetiscoaaDepartment of Mathematics, University of the Basque Country, Barrio Sarriena s/n, 48940, Leioa, Spain,
jonasier.barcena@ehu.eus

Abstract
In this talk we prove the observability of parabolic equations on networks with loops, and use this to obtain results
related with inverse problems. We show that the observability of the entire network can be achieved under certain
hypotheses about the position of the observation domain. This is done using a novel Carleman inequality. The maindifficulty we tackle, due to the existence of loops, is to avoid entering into a circular fallacy, notably in the constructionof the auxiliary function for the Carleman inequality. This is overcome as the hypotheses allow setting from which end
the heat on the edge is observed on the edges with empty intersection with the observation domain.

57

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

ADAPTIVE FINITE ELEMENT METHOD FOR SOLUTION OF INVERSE AND ILLPOSED PROBLEMS

Larisa Beilina,
Chalmers University of Technology and University of Gothenburg, SE-41296, Gothenburg, Sweden,larisa.beilina@chalmers.se

Abstract
An adaptive finite element method for application to solution of coefficient inverse and some ill-posed
problems will be discussed. These problems usually are formulated as optimal control problems for minimization of the
Tikhonov's regularization functional [1].
The main idea of an adaptive finite element method is following: it allows to obtain good accuracy of thecomputed solution of inverse or ill-posed problem via local adaptive mesh refinements. To be able use this idea, theTikhonov functional is minimized on a sequence of locally refined finite element meshes. These meshes are refined
based on the posteriori error estimates in the obtained finite element reconstruction or in the error in the Tikhonov's
functional. We will outline main steps in derivation of a posteriori error estimates. Based on these estimates, adaptive
finite element algorithms will be formulated and their work will be demonstrated on the real-life problem of monitoringof hyperthermia [2].

References
1. L. Beilina, M. V. Klibanov, Approximate global convergence and adaptivity for coefficient inverse problems,
Springer, New-York, (2012).
2. M. G. Aram, L. Beilina, H. D. Trefna, Microwave Thermometry with Potential Application in non-invasive
monitoring of hyperthermia. J. Inverse Ill‐Posed Probl. 1 (2020). DOI: 10.1515/jiip-2020-0102 in Non-invasive
Monitoring of Hyperthermia, Journal of Inverse and Ill-posed problems, 2020.

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The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

MULTILEVEL CONTROL

Umberto Biccaria,b, Enrique Zuazuaa,c,d , Carlos Esteve-Yagüee and Deyviss Jesús Oroya-VillaltabaChair of Computational Mathematics, Fundación Deusto, Av. Universidades 24, 48007 Bilbao, BasqueCountry, Spain, umberto.biccari @ deusto.es;bUniversidad de Deusto, Av. Universidades 24, 48007 Bilbao, Basque Country, Spain;c
Chair for Dynamics, Control and Numerics, Alexander von Humboldt-Professorship, Dept. of Data
Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058, Erlangen, Germany;d
Depto. de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain;e
Faculty of Mathematics, Cambridge University, U.K.

Abstract
We discuss the multilevel control problem for dynamical systems, consisting in designing a piecewise constantcontrol function taking values in a finite-dimensional set. We start by providing a complete characterization ofmultilevel controls through an optimal control approach, based on the minimization of a suitable cost functional. In thismanner we build optimal multilevel controls and characterize the time needed for a given ensemble of levels to assure
the controllability of the system [1]. In a second moment, as a practical application of our results, we employ themultilevel control strategy to efficiently solve the Selective Harmonics Modulation problem in power electronics
engineering [2].

References
1. U. Biccari and E. Zuazua, Multilevel control by duality, preprint, arXiv:2109.02346
2. U. Biccari, C. Esteve-Yagüe and D. J. Oroya-Villalta, Multilevel Selective Harmonic Modulation via OptimalControl, preprint, arXiv:2103.10266

59

The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

NUMERICAL RECONSTRUCTION OF THE CONDUCTIVITY IN THE 3-D CALDERON
PROBLEM USING THE BORN APPROXIMATION
J.A. Barcelóa, C. Castroa, F. Maciàa and C. Meroñoaa
M2ASAI. Universidad Politécnica de Madrid, ETSI Caminos, c. Profesor Aranguren s/n, 28040, Madrid, Spain,
juanantonio.barcelo@upm.es, carlos.castro@upm.es, fabricio.macia@upm.es, cjmerono@upm.es;

Abstract
Reconstruction in the three-dimensional Calderon inverse conductivity problem can be reduced to the study ofthe inverse boundary problem for Schrödinger operators −∆ + q. A suitable linearization of the inverse problem allows
to define the so-called Born approximation. As described in [1], this provides a good reconstruction only for the highfrequencies of q. In a recent work [2], we established a new formula relating this Born approximation with the Dirichlet
to Neumann map and the momenta of q. Based on this formula, we give a new convergent iterative algorithm that
improves the numerical reconstruction given by the Born approximation. We illustrate the process with numericalexperiments for radial and non-radial potentials.

References
1. J. Bikowski, K. Knudsen, and J. L. Mueller. Direct numerical reconstruction of conductivities in three dimensions
using scattering transforms. Inverse Problems, 27(1) 015002, 19pp, (2011).
2. J.A. Barceló, C. Castro, F. Macià and C. Meroño, The Born approximation in the three-dimensional Calderon
problem, arXiv:2109.06607 [math.AP] (2021).

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The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

TRANSCRANIAL DIRECT CURRENT STIMULATION: ESTIMATION OF INDIVIDUALLYTARGETED USING THE HEAD GEOMETRY

Jairo Rocha de Faria
Department of Computer Systems, Federal University of Paraiba, João Pessoa, PB, Brazil, jairo@ci.ufpb.br

Abstract
Electrical stimulation to modify brain function has been studied since Bartholow's pioneering work [1] in 1874.
Nowadays, among the many modern non-invasive neurostimulation techniques, transcranial direct current stimulation(tDCS) stands out, which has been applied to reduce the symptoms of post-stroke motor and language deficits,
associated with depression, chronic pain, memory loss and tinnitus. The required equipment is simple, versatile, easy
applied and relatively low in cost. Furthermore, its adverse effects are rare, transient, and well-tolerated, in general,
ranging from itching, tingling, and burning sensations to redness on the stimulated areas, headaches, nausea, andinsomnia [2].
However, the specialized literature indicates that results strongly depend on the adopted protocol [3] (such as
electrode positions, duration of stimulation, number of sessions, intensity of current, etc.) and the head and brain’s
anatomy [4], indicating a need to shift from a one-size-fits-all pacing approach to an individualized induced e-fieldstrength.
The associated inverse problem, therefore, is given a prescribed electric field distribution, defining the electrodeplacement and the required current intensity, accounting for head and brain configurations.
In this work we used the open-source software SimNIBS (Simulation of Non-invasive Brain Stimulation) in order
to consider a realistic finite element head model. Then, we estimated the electrode placement taking into account the
topological derivative [5] for a suitable shape functional, considering several cranial geometries. Finally, we performed
the hierarchical clustering dendrogram, associated to the single-linkage clustering [6], in order to estimate
individualized tDCS parameters.

References

1. R. Bartholow, Art. I.—Experimental Investigations into the Functions of the Human Brain. The American Journal of
the Medical Sciences, 66, 305-313 (1874).
2. M. Bikson, P. Grossman, C. Thomas, A. L. Zannou, J. Jiang, T. Adnan, A. P. Mourdoukoutas, G. Kronberg, D.
Truong, P. Boggio, A. R. Brunoni, L. Charvet, F. Fregni, B. Fritsch, B. Gillick, R. H. Hamilton, B. M. Hampstead, R.
Jankord, A. Kirton, H. Knotkova, D. Liebetanz, A. Liu, C. Loo, M. A. Nitsche, J. Reis, J. D. Richardson, A.
Rotenberg, P. E. Turkeltaub, A. J. Woods. Safety of Transcranial Direct Current Stimulation: Evidence Based Update2016. Brain Stimulation 9(5):64-661 (2016).
3. A. J. Woods, A. Antal, M. Bikson, P. S. Boggio, A. R. Brunoni, P. Celnik, L. G. Cohen, F. Fregni, C. S. Herrmann,
E. S. Kappenman, H. Knotkova, D. Liebetanz, C. Miniussi, P. C. Miranda, W. Paulus, A. Priori, D. Reato, C. Stagg,
N. Wenderoth, & M. A. Nitsche. A technical guide to tDCS, and related non-invasive brain stimulation tools. Clinicalneurophysiology: official journal of the International Federation of Clinical Neurophysiology, 127(2), 1031-1048(2016).
4. A. Guerra, V. López-Alonso, B. Cheeran, & A. Suppa. Solutions for managing variability in non-invasive brain
stimulation studies. Neuroscience letters, 719, 133332 (2020).
5. A. A. Novotny, J. Sokolowski. An Introduction to the Topological Derivative Method. SpringerBriefs inMathematics, Springer International Publishing, DOI 10.1007/978-3-030-36915-6 (2020).
6. R. Rabadan & A. Blumberg, Topological Data Analysis for Genomics and Evolution: Topology in Biology.
Cambridge: Cambridge University Press. (2019)

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The 10th International Conference “Inverse Problems: Modeling and Simulation”(IPMS-2028)
Malta, May 22 – 28, 2022

A RESISTANCE METHOD FOR A FLUID-OBSTACLE INVERSE PROBLEM

A. Ossesa, J. Aguayoa,b and C. Bertoglioba
DIM-CMM, U. de Chile, Beauchef 851, Santiago, Chile, axosses@dim.uchile.cl; jaguayo@dim.uchile.cl;bBernoulli Institute, University of. Gröningen, The Netherlands, c.a.bertoglio@rug.nl;

Abstract
We present an obstacle inverse problem in fluids. We analyze it from the point of view of recovering theboundary obstacle from velocity measurements and unknown pressure. First we show how to approximate thegeometrical inverse problem with a parameter identification problem and then we establish some uniqueness andstability results using Carleman inequalities.
The problem is motivated from cardiac valve detection using MRI-4D flow measurements.

References
1. J. Aguayo, C. Bertoglio and A. Osses. A Distributed Resistance Inverse Method for Flow Obstacle Identificationfrom Internal Velocity Measurements. Inverse Problems, 37(2), 025010 (2021). doi: 10.1088/1361-6420/abced8

62

MINISYMPOSIUM
M6: Generalized Radon Transforms and Applications
Organizers:
Fedor Goncharov, CMAP, Ecole Polytechnique, France, fedor.goncharov.ol@gmail.com
Mai K. Nguyen-Verger, University of Cergy-Pontoise, France, mai.nguyen-verger@u-cergy.fr
T. T. Truong, University of Cergy-Pontoise, France, truong@u-cergy.fr
The aim of this mini-symposium is to gather scientists working in the field of imaging inverse
problems soluble by generalized Radon transforms. There has been many advances by various groups overthe last decade. It is of interest to provide an overview of progress in this research field as well as related
fields. Applications range from imaging in seismic, medical diagnostics, industrial non-destructive
evaluation and control to detection-identification in monitoring nuclear safety, homeland security andecological survey.
It is expected that participants would have the opportunity to meet each other, exchange ideas andpossibly start collaborative work.

63

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

THE MODULO RADON TRANSFORMS AND ITS INVERSION
M. Beckmanna , F. Krahmerib and A. BhandaricaDepartment of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany,
matthias.beckmann@uni-hamburg.debDepartment of Mathematics, Technical University of Munich, Garching 85747, Germany,
felix.krahmer@tum.decDepartment of Electrical and Electronic Engineering, Imperial College London, SW72AZ, UK,a.bhandari@imperial.ac.uk

Abstract
In this paper, we introduce the Modulo Radon Transform (MRT) which is complimented by an inversion
algorithm. The MRT generalizes the conventional Radon Transform and is obtained via computing modulo of the line
integral of a two-dimensional function at a given angle. Since the modulo operation has an aliasing effect on the range
of a function, the recorded MRT sinograms are always bounded, thus avoiding information loss arising from saturation
or clipping effects. This paves a new pathway for imaging applications such as high dynamic range computed
tomography. By capitalizing on the recent results on Unlimited Sensing architecture, we prove that the Modulo Radon
Transform can be inverted when the resultant (discrete/continuous) measurements map to a bandlimited function.
Thus, the MRT leads to new possibilities for both conceptualization of inversion algorithms as well as
development of new hardware

64

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STRUCTURED ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
Maimouna Bocoum1, Remi Carloni Gertosio2, Jean-Luc Gennisson2, Claude Comtat2, Marina Pierucci
Filipovic2 and Francois Ramaz1
1
Institut Langevin, Ondes et Images - ESPCI Paris, PSL Research University, CNRS UMR 7587,
INSERM U979, Universite Paris VI Pierre et Marie Curie, 1 rue Jussieu, 75005 Paris, France
2
BIOMAPS, laboratoire d'imagerie biomedicale multi-modale a Paris-Saclay, Universite Paris-Saclay,
CNRS,INSERM, CEA, 4 Place du Generale Leclerc, 91401 Orsay, France
maimouna.bocoum@espci.fr

Abstract
Ultrasound-modulated optical tomography (UOT) is an imaging technique which couples light and ultrasound(US) in order to perform in-depth optical imaging of highly scattering media [4, 8, 7, 9]. A potential application ismillimeter to centimeter-deep imaging of living tissues for medical diagnosis. Traditional images are obtained using
MHz-range few-cycle Focused Waves (FW) as illustrated in Fig 1(a). By instead using Plane-Waves (PW) to tag
photons, we can reconstruct an image using the filtered backprojection method [2, 1, 5]. Because of angular limitation
however, the measurement suffers from a drastic loss in lateral spatial resolution. Here, we present a new structured
ultrasonic plane wave UOT method that allows partial recovery of the resolution. We will introduce a generalizationof the Fourier Slice Theorem along with a generalized filtered backprojection formalism.
The method is successfully tested on simulated and experimental data [3].

Fig. 1 : Experimental images of two absorbing inclusions obtained using respectively FW(a), PW(b),and SWultrasonic waves(c). All images can be found with extensive descriptions in [3].
PW imaging consists in tagging photons using multiple a few-cycle US pulses emitted at different angles asrepresented in 1(b). As illustrated by the image below, the reconstruction quality in the lateral direction is poor because
the angular span is limited by the acoustic probe directivity. PW-UOT remains nevertheless a powerfull alternative toFW when fast acquisition speed is required [6].

References[1] Dale L Bailey, Michael N Maisey, David W Townsend, and Peter E Valk. Positron emission tomography. Springer,
2005.
[2] Jose Baruchel, Jean-Yves Buffiere, and Eric Maire. X-ray tomography in material science. 2000.
[3] Maimouna Bocoum, Jean-Luc Gennisson, Jean-Baptiste Laudereau, Anne Louchet-Chauvet, Jean-Michel Tualle,and Francois Ramaz. Structured ultrasound-modulated optical tomography. Applied Optics, 58(8):1933-1940, 2019.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

[4] Maimouna Bocoum, Jean Luc Gennisson, Caroline Venet, Mingjun Chi, Paul Michael Petersen, Alexander A
Grabar, and Franccois Ramaz. Two-color interpolation of the absorption response for quantitative acousto-optic
imaging. Optics Letters, 43(3):399-402, 2018.
[5] Joachim Frank. Electron tomography: methods for three-dimensional visualization of structures in the cell. Springer
Science & Business Media, 2008.
[6] Jean-Baptiste Laudereau, Alexander A Grabar, Mickael Tanter, Jean-Luc Gennisson, and Francois Ramaz. Ultrafast
acousto-optic imaging with ultrasonic plane waves. Optics Express, 24(4):3774- 3789, 2016.
[7] Fay A Marks, Harold W Tomlinson, and Glen W Brooksby. A comprehensive approach to breast cancer detection
using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination. In
Proc. SPIE, volume 1888, pages 500-510, 1993.
[8] Michael S Patterson, Britton Chance, and Brian C Wilson. Time resolved reectance and transmittance for thenoninvasive measurement of tissue optical properties. Applied Optics, 28(12):2331-2336, 1989.
[9] Lihong Wang, Steven L Jacques, and Xuemei Zhao. Continuous-wave ultrasonic modulation of scattered laser lightto image objects in turbid media. Optics Letters, 20(6):629-631, 1995.

66

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON RANGE CONDITION OF THE X-RAY TRANSFORM IN R n
Aleksander Denisiuk
University of Warmia and Mazury in Olsztyn, Poland,denisiuk@matman.uwm.edu.pl

Abstract
Consider the problem of the range description of the tensor x -ray transform. Specific issue is to minimize thenumber of differential equations that describe the range.
In this study a geometrical interpretation of the range condition and related John differential operator is given. As acorollary, it is proved that the range of the m-tensor x-ray transform in R
differential equations.

67n n  m  2
 m 1 

can be described by The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RAY TRANSFORM PROBLEMS ARISING FROM SEISMOLOGY ON MARS
Joonas Ilmavirta
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014

Abstract
We discuss recent uniqueness results in the mathematical theory of inverse problems and their relation toseismic investigations of Mars using NASA's InSight lander. Constrained by severely limited data, one must be carefulin drawing conclusions from measurements.
We discuss some of the mathematical inverse problems, especially in ray tomography, that arise from thissetting.

68

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SPECTRAL COMPTON CAMERA IMAGING
OF POLYCHROMATIC GAMMA-RAY SOURCES
Voichita Maxim
University of Lyon, INSA‐Lyon, Universite Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm,
CREATIS UMR 5220, U1206, F‐69XXX, LYON, France, voichita.maxim@creatis.insa-lyon.fr

Abstract
In this work, we investigate spectral image reconstruction for Compton camera gamma-ray imaging. The
forward model is defined to take into account incomplete absorption and to accurately describe the acquisition process
at each energy. Total variation regularization can be added to compensate for low dose.
We present a novel pre-conditioned algorithm that allows an important reduction of the number of TV
iterations. This algorithm can be applied to other tomography problems where the sensitivity matrix has importantvariations through the reconstructed volume, for instance in CBCT.

69

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOME GENERALIZED RADON TRANSFORMS INSPIRED FROM IMAGINGTECHNOLOGY
Mai K. Nguyen
University of Cergy-Pontoise, France, mai.nguyen-verger@u-cergy.fr

Abstract
We discuss some Generalized Radon Tranforms (GRT) which arise from various detection technologies ofionizing radiation, electromagnetic radiation or sound waves. They encompass the great majority of the relevant Radontransforms in imaging applications, namely the reconstruction of hidden structures of interest. Primary (or undeviated)radiation detection involves the well known classical Radon transform in Computed Tomography. However detectionof transmitted radiation with Compton scattering over illuminated objects, at chosen energies leads to generalized
Radon transforms on various families of Circular Arcs in the so-called Compton Scatter Tomography.
In addition, Emission tomography modalities such as SPECT or PET requires anew the classical Radontransform but Compton Scatter Emission Imaging introduces Conical Radon transform in three dimensions and the Vline Radon transform in two dimensions. Photo-Acoustic-Tomography (PAT) (resp. SAR) imaging uses Radontransforms on Spheres (or Circles) centered on a fixed circle (resp. a fixed line).
The main properties of the inversion formulas of these GRTs and their illustrating numerical simulations arepresented to support their abilities for useful imaging applications (medical diagnostics, industrial non-destructive
evaluation, security, cultural heritage object imaging, etc.).
This is a joint work with Cécilia Tarpau, Javier Cebeiro, T.T. Truong.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AUGMENTED NETT REGULARIZATION OF INVERSE PROBLEMS
Daniel Obmanna, Linh Nguyenb, Johannes Schwaba, Markus HaltmeieraaDepartment of Mathematics, University of Innsbruck, daniel.obmann@uibk.ac.atb
Department of Mathematics, University of Idaho, lnguyen@uidaho.edu

Abstract
We propose aNETT (augmented NETwork Tikhonov) regularization as a novel data-driven reconstruction
framework for solving inverse problems. An encoder-decoder type network defines a regularizer consisting of a penalty
term that enforces regularity in the encoder domain, augmented by a penalty that penalizes the distance to the data
manifold. We present a rigorous convergence analysis including stability estimates and convergence rates. For that
purpose, we prove the coercivity of the regularizer used without requiring explicit coercivity assumptions for the
networks involved. We propose a possible realization together with a network architecture and a modular training
strategy. Applications to the inversion of the Radon transform are presented. This talk is based on [1].

References[1] Obmann, D., Nguyen, L., Schwab, J. and Haltmeier M., 2021, Augmented NETT Regularization of Inverse
Problems. https://arxiv.org/abs/1908.03006

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

COMPTON SCATTER TOMOGRAPHY: FROM ALGEBRAIC AND ITERATIVERECONSTRUCTION TO MACHINE LEARNING
S. Pistoriusa, G. Fontainea, H.Y. Sun, G. Zang, H.W. Sun and T. ChigvinadzeaDepartment of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada,
stephen.pistorius@umanitoba.ca

Abstract
Introduction: X-ray and gamma-ray photons deposit energy and scatter when they interact with human tissue.
Scattered radiation has historically been considered to be noise, contributing dose to the patient, but providing no useful
information. Traditional imaging systems such as Computed Tomography (CT) and Positron Emission Tomography(PET) provide valuable diagnostic information but contribute to a growing population dose. With the increased
availability of higher energy-resolution detectors and advanced Machine Learning approaches, scatter imaging is
becoming feasible. This work will summarize the results of our CT and PET scatter-imaging research. I will illustrate
the theory and will describe how the scatter-imaging techniques improve our ability to diagnose cancer without thedose or cost penalties associated with multi-modality scanning, can provide attenuation corrections, and give ananatomical context to PET, which is otherwise a functional imaging modality.
Methods and Materials: Our initial work in CT used a refined version of the backprojection algorithm overarcs to demonstrate that by measuring the energy and position of detected photons, we can reconstruct images withsignificantly fewer projections. For scatter-reconstruction in PET, generalized Maximum-Likelihood ExpectationMaximization (G*S-MLEM) reconstruction algorithms were developed, which are capable of reconstructing single anddual-scattered photons. The scattering angle, calculated from the energy of each photon using the Compton equation,
was used to define two circular arcs (TCA) that encompass the annihilation position. In the zero-degree scattering angle
limit, the scattered coincidences approach the true coincidence, and hence trues are considered a subset of scatteredcoincidences. To avoid overcorrecting for scattered coincidences, the attenuation coefficient was calculated byintegrating the differential Klein-Nishina cross-section over a restricted-energy range, accounting only for scatteredphotons that were not detected. Phantoms containing cold and hot regions with various activities were simulated using
the GATE platform. Energy resolutions of 5% to 20% were used to blur the simulated events, with scatter fractions
from 10 to 40%. Images were reconstructed using different algorithms with a 350-650 keV energy window and the
proposed restricted attenuation correction. Convolutional Neural Networks have been trained and tested using 120,000
PET images generated using simulated analytical PET images and with Monte Carlo simulation. The Monte Carlosimulation was used to obtain the energy distribution of the scattered photons, and transfer learning has been used tominimize the training data requirements.
Results and Conclusions: We have developed reconstruction algorithms capable of reconstructing CT and
PET images using scattered photon coincidences. These approaches improve contrast, decreases noise, and eliminates
the need for scatter corrections. Using scattered photons lowers the data required for equivalent image quality, reducesscan time, injected dose, and radiation burden. Images generated using single scattered coincidences demonstrate that
only 70% of annihilation positions were correctly identified, while the generalized dual-scatter GDS-MLEM algorithm
encompassed 98% of source positions. The GDS-MLEM has the highest sensitivity, improved contrast recovery
coefficient, and reduced noise by 7.6% to 13.2% and 12.4% to 24.0%, respectively. The revised attenuation correction
method facilitates dynamic selection of an energy window to provide optimum PET images. The GDS-MLEM
approach is less sensitive to energy resolution and shows promise if detector energy resolutions of 12% can beachieved. While this algorithm is an improvement over our earlier approaches, our AI-based CNN deep-learning
reconstruction approaches promise to provide still better quality images that are less sensitive to detector energy
resolution.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FOURIER RECONSTRUCTION IN DIFFRACTION TOMOGRAPHY
Michael Quellmalz
TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany, quellmalz@math.tu-berlin.de

Abstract
We study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a rigid
particle rotating in a trap created by acoustic or optical forces. Under the influence of the inhomogeneous forces, theparticle carries out a time-dependent smooth, but irregular motion. The rotation axis is not fixed, but continuously
undergoes some variations, and the rotation angles are not equally spaced, which is in contrast to standard tomographic
reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, andthat the particle’s scattering potential is compatible with making the first order Born or Rytov approximation. By the
Fourier diffraction theorem, the measurements of the scattered wave are related to the Fourier transform of thescattering potential on an irregular grid. We derive novel backpropagation formulae for the reconstruction of thescattering potential, which depends on the refractive index inside the object, taking its complicated motion into account.
This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform [2,3].
Furthermore, we consider the case of missing phase information, since often only intensity measurements areavailable in practice. We propose a new reconstruction approach [1] for ODT with unknown phase information,utilizing a hybrid input-output scheme with TV regularization. The so-obtained 2D and 3D reconstructions are evencomparable to the ones with known phase.

References
1. R. Beinert and M. Quellmalz, Total variation-based phase retrieval for diffraction tomography, Arxiv preprint(2022).
2. F. Faucher, C. Kirisits, M. Quellmalz, O. Scherzer, E. Setterqvist, Diffraction tomography, Fourierreconstruction, and full waveform inversion, Handbook of Mathematical Models and Algorithms in Computer Vision
and Imaging, accepted (2022).
3. C. Kirisits, M. Quellmalz, M. Ritsch-Marte, O. Scherzer, E. Setterqvist, G. Steidl, Fourier reconstruction for
diffraction tomography of an object rotated into arbitrary orientations, Inverse Problems. 37, 115002 (2021).

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

PERIODIC VIEW OF SOME CLASSICAL OPERATORS ASSOCIATED TO RADON
TRANSFORMS
Jesse Railo
University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland,jesse.t.railo@jyu.fi

Abstract
We discuss our recent results and ongoing work on the periodic Radon transforms. Certain dense partial dataof the usual Radon transform can be mapped into periodic Radon transform data, which motivates studying the periodic
counterpart of the Radon transform. Geometrically in two dimensions, the periodic Radon transform is the geodesic raytransform over closed geodesics of the flat torus. We compare some classical operators and results in both settings suchas the Fourier slice theorem, the (filtered) backprojection, adjoint, normal and inverse operators, and their mapping
properties. The periodic Radon transforms enjoy some remarkable properties which give simple reconstruction
formulas and sometimes a bit surprising results that do not resemble their classical counterparts. One can obtainstability and regularization results in low regularity when data spaces are equipped with suitable (weighted) Sobolev or
Bessel type structures.
This talk is based on series of works by Joonas Ilmavirta (Jyväskylä), Olli Koskela (Tampere), and the author;
arXiv:1402.6209, arXiv:1906.05046, arXiv:1909.00495.

74

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DIFFRACTIVE TENSOR FIELD TOMOGRAPHY AS AN INVERSE PROBLEM FOR A
TRANSPORT EQUATION
Lukas Vierus
Saarland University, Germany
vierus@num.uni-sb.de

Abstract
We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting andabsorbing medium as inverse source problem for the associated transport equation. Following Fermat's principle the
Riemannian metric in the considered domain is generated by the refractive index of the medium. There is wealth ofresults for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static euclidean
setting, whereas there are only few inversion formulas and algorithms existing for general Riemannian metrics andtime-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source
problem for a transport equation where the ray transform serves as given boundary data. We prove that this result
extends to the dynamic case. Interpreting dynamic tensor tomography as inverse source problem represents a holistic
approach in this field. To guarantee that the forward mappings are well-defined, it is necessary to prove existence anduniqueness for the underlying transport equations. Unfortunately the bilinear forms of the associated weak formulations
do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence inappropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows
only small variations of the refractive index.
Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity
term turns to zero75

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FROM ANALYSIS TO ITERATIVE ALGORITHMS: A LINEAR SYSTEMSPERSPECTIVE ON THE BROKEN RAY TRANSFORM
Michael R. Walker IIa and Joseph A. O’SullivanbaSandia National Laboratories, Albuquerque, NM, USA, mrwalke@sandia.govbPreston M. Green Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis,
MO, USA, jao@wustl.edu

Abstract
Analysis of the BRT has led to elegant inversion formulas for bounded images. A linear systems perspective
provides a common framework for summarizing prior results and the challenges extending them to sampled data. We
recognize two issues with the BRT: unbounded support of the data, and nontrivial nullspace of the forward operator.
Addressing these challenges enables computationally efficient inversion formulas in the frequency domain for arbitrary
ray directions. Further, our results yield fast implementations of the forward and backward transforms which arerequired for a broad class of iterative reconstruction algorithms. Recently we have used iterative algorithms to jointly
estimate scatter density images and attenuation images from noisy data. Our results illustrate the sensitivity of joint
image reconstruction to problem scaling and quantify the benefits of incorporating transmission measurements andadditional scatter angles.

References
1. M. R. Walker and J. A. O’Sullivan, "Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken
Ray Transform Data," in IEEE Transactions on Computational Imaging, vol. 7, pp. 361-374, 2021
2. M. R. Walker and J. A. O’Sullivan, “The broken ray transform: additional properties and new inversion formula,” inInverse Problems, vol. 25, 201976

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A JOINT RECONSTRUCTION AND LAMBDA TOMOGRAPHY REGULARIZATIONTECHNIQUE FOR ENERGY RESERVED X-RAY IMAGINGJames Webber
Department of Electrical and Computer Engineering, Tufts University, Medford, MA USA
E-Mail: James.Webber@tufts.edu

Abstract
Here we present new joint reconstruction and regularization techniques inspired by ideas in microlocal
analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coeffcient and electron densityfrom X-ray transmission (i.e., X-ray CT) and backscattered data (assumed to be primarily Compton scattered). To
demonstrate our theory and reconstruction methods, we consider the “parallel line segment” acquisition geometry ofWebber and Miller (Compton scattering tomography in translational geometries. Inverse Problems, 36, no. 2 (2020):
025007), which is motivated by system architectures currently under development for airport security screening.
We first present a novel microlocal analysis of the parallel line geometry which explains the nature of image
artefacts when the attenuation coeffcient and electron density are reconstructed separately. We next introduce a new
joint reconstruction scheme for low effective Z (atomic number) imaging ( Z  20 ) characterized by aregularization strategy whose structure is derived from lambda tomography principles and motivated directly by the
microlocal analytic results.
Finally we show the effectiveness of our method in combating noise and image artefacts on simulated
phantoms.

77

MINISYMPOSIUM
M7: Recent Advances in Inverse Problems of Time-Harmonic Wave Propagation
Organizers:
Shixu Meng, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China,
shixumeng@amss.ac.cn
Roman Novikov, Ecole Polytechnique, France, roman.novikov@polytechnique.edu
Many developing and emerging fields in physics, astronomy and engineering are rich sources of
inverse problems of time-harmonic wave propagation. The aim of this minisymposium is to bring together
experienced and young researchers interested in time-harmonic inverse problems and to discuss recent
advances, challenges and applications of this discipline. Among other things, we will focus on uniqueness,reconstruction algorithms, problems with partial data and simulations.

78

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

EIGENVALUE PROBLEM FOR A SCATTERER WITH A CONDUCTIVE BOUNDARY

Isaac Harris
Department of Mathematics, Purdue University, USA
harri814@purdue.edu

Abstract
In this talk, we will investigate the inverse acoustic scattering problem associated with an inhomogeneous
media with a conductive boundary. We consider the corresponding classical transmission eigenvalue problem as wellas the zero-index eigenvalue problem. This is a new class of eigenvalue problem that is not elliptic, not self-adjoint, andnon-linear, which gives the possibility of complex eigenvalues.
We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as proveexistence and discreteness for the case of an absorbing media.

79The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

UNIQUENESS RESULT FOR RANDOM INVERSE SOURCE PROBLEMS ANDAPPLICATIONS TO HELIOSEISMIC HOLOGRAPHY

Thorsten Hohage
University of Göttingen, Germany, hohage@math.uni-goettingen.de
Max Planck Institute for Solar Systems Research, Germany

Abstract
We consider the problem of reconstructing the intensity of an uncorrelated random source from measurementsof solutions to a Helmholtz-type equation at some distance to the source. We prove that under mild assumptions theintensity is uniquely determined by correlations of the random fields. In contrast, it is well that corresponding
deterministic inverse source problems are highly non-unique, i.e. it is not possible to determine individual source
realizations.
Such problems appear in locating noise source in aircrafts, trains or cars as well as in local helioseismology.
We also discuss some well-established algorithms for reconstructing source intensities in a aeroacoustics andhelioseismic holography as well as some new algorithms and show numerical results.

80

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A DIRECT AND INVERSE SCATTERING PROBLEM FOR A LOCALLY PERTURBEDPERIODIC STRUCTURE

Andreas Kirsch
Department of Mathematics, KIT, 76128 Karlsruhe, Germany
andreas.kirsch@kit.edu

Abstract
In this talk we will study the scattering of time-harmonic point sources (or other sources of compact support)
in some domain of the plane which is a local perturbation of a domain D where D is periodic with respect to thehorizontal axis and bounded with respect to the vertical axis.
The main part of the talk is concerned with the discussion of a suitable radiation condition for this scattering
problem and present results for uniqueness and existence. Then we turn to the inverse problem to determine propertiesof the perturbation from the measured fields and prove uniqueness of the inverse problem.

81

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INTERIOR TRANSMISSION EIGENVALUES FOR FRACTAL DOMAINS

Andreas Kleefeld
Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52425 Jülich, Germany
a.kleefeld@fz-juelich.de

Abstract
Interior transmission eigenvalues arise in the study of reconstruction algorithm for inverse scattering problems. The
numerical calculation of those is a challenging task due to the fact that the problem is non-linear, non-elliptic, and nonself-adjoint (see for example [1] and [2]). It gets even more complicated when the two-dimensional domain underconsideration has a fractal boundary. It is shown that we are able to successfully compute interior transmission
eigenvalues for such domains.
Interesting observations and conjectures are made which hopefully stimulate the community to pursue further
research in this direction.

References
1. A. Kirsch and A. Lechleiter, The inside-outside duality method for scattering problems by inhomogeneous media,
Inverse Problems, 29, 104011 (2013).
2. A. Kleefeld, A numerical method to compute interior transmission eigenvalues, Inverse Problems, 29, 104012(2013).

82

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SINGLE MODE MULTI-FREQUENCY FACTORIZATION METHOD FOR THE INVERSE
SOURCE PROBLEM IN ACOUSTIC WAVEGUIDES

Shixu Meng
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China
shixumeng@amss.ac.cn

Abstract
This talk addresses the inverse source problem with a single propagating mode at multiple frequencies in an
acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended
sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the
space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over thefrequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation,a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of therange support. A related factorization-based sampling method is also discussed. These sampling methods are shown tobe capable of imaging the range support of the source.
Numerical examples are provided to illustrate the performance of the sampling methods, including an example toimage a complete sound-soft block.

83

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

QUANTITATIVE PASSIVE IMAGING BY ITERATIVE HELIOSEISMIC HOLOGRAPHY

B. Müllera, T. Hohageb, L. Gizona, D. FournieraaMax-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany,
muellerb@mps.mpg.de;b
Institut für Numerische und Angewandte Mathematik, Lotzstr. 16-18, 37083 Göttingen, Germany

Abstract
In local helioseismology one studies solar oscillations at the solar surface (e.g. line-of-sight velocities)
in order to learn about subsurface quantities (sound speed , density , flows , wave attenuation ) [1]. Due to thestochastic nature of solar oscillations the input data consists of cross-correlationsat any point pairatthe solar surface. In the frequency domain the cross-correlation takes the form.
Acoustic wave propagation in the Sun can be described by a simplified scalar equation in the frequency domain
,

,

(1)

where is the stochastic source term. Under the reasonable assumption of equipartition of the source power, the crosscovariancecan be assumed to be proportional to[2].
In this presentation, we will address the Inverse Problem of reconstructing(in particular
axisymmetric subsurface flows) from observations of cross-correlations at the solar surface. The cross-correlation data
is a very large five-dimensional data set, which is unfeasible to store and can not be inverted directly. Therefore some apriori averaging in spatial or frequency direction is necessary. Traditional approaches reduce the cross-correlation to asmaller number of physically interpretable measurements (e.g. travel times) with acceptable signal-to-noise ratio.
Helioseismic holography consists in first backpropagating the data and afterwards locally cross-correlating the
backpropagated wavefield in order to image the solar interior at a previously defined target location. This way
holography uses the whole cross-correlation data implicitly without the need to compute the cross-correlation
explicitly. Nevertheless seismic holography only provides feature maps and is no quantitative regularization method.
We will introduce an appropriate forward operator mapping to the cross-covariance at the surface [3] in order to solve
the inverse problem using the Iteratively Regularized Gauss Newton Method with Conjugate Gradient inner iterations.
The holographic backpropagation can be interpreted as the adjoint of the Frèchet derivative of the forward problem,
such that the first iteration coincides with traditional holographic image intensity. This iterative approach is motivatedby 3 main advantages compared to traditional approaches (like time-distance helioseismology):
1) The whole input cross-correlation data can be used without the loss of information caused by a priori reduction
steps.
2) Helioseismology can be extended to nonlinear inverse problems, stepping forward to full-waveform inversions.
3) The gradient and the adjoint can be obtained by solving similar equations to (1) with modified right hand side,which can be implemented in a computational efficient way.

References
1. T. L. Duvall, S. M. Jefferies, J. W. Harvey and M. A. Pomerantz, Time-Distance Helioseismology, Nature, 362, 430432 (1993).
2. L. Gizon, H. Barucq, M. Durufle, C.S. Hanson, M. Leguebe, A.C. Birch, J. Chabassier, D. Fournier, T. Hohage, E.
Papini, Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models withflows, A&A, 600, A35 (2017).
3. T. Hohage, H. Raumer, C. Spehr, Uniqueness of an inverse source problem in experimental aeroacoustics , Inverse
Problems, 36 (2020).

84

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FORMULAS FOR PHASE RECOVERING FROM PHASELESS SCATTERING DATA

Roman G. Novikova,baCMAP, CNRS, ´Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, Franceb
IEPT RAS, 117997 Moscow, Russia; e-mail: novikov@cmap.polytechnique.fr

Abstract
We present formulas for phase recovering from appropriate monochromatic phaseless scattering data indimension d  1 . In particular, these formulas reduce phaseless inverse scattering to standard inverse scattering withphase information.
This talk is based, in particular, on works [1-4].

References[1] R.G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian Journal of Mathematical andComputer Applications 3(1), 63-69 (2015)[2] R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin desSciences Mathmatiques 139(8), 923-936 (2015)[3] R.G. Novikov, Multipoint formulas for phase recovering from phaseless scattering data, J. Geom. Anal.,
doi:10.1007/s12220-019-00329-6[4] R.G. Novikov, V.N. Sivkin, Error estimates for phase recovering from phaseless scattering data, Eurasian Journalof Mathematical and Computer Applications 8(1), (2020)

85

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON COMPLEX TRANSMISSION EIGENVALUES
L. Pieroneka, A. KleefeldbaKarlsruhe Institute of Technology, 76131 Karlsruhe, Germany, lukas.pieronek@kit.edu;bJuelich Supercomputing Centre, 52425 Juelich, Germany, a.kleefeld@fz-juelich.de

Abstract
Transmission eigenvalues arise in the study of inverse scattering problems for time-harmonic and penetratingwaves. They are closely connected to non-scattering wavenumbers and depend on the refractive index of some
practically unknown scattering domain which one aims to reconstruct. Sampling methods, which are designed for thispurpose and exploit far field measurements of monochromatic incident waves, may fail correspondingly. On the other
hand, given a fixed scattering domain such as a ball, it has been shown that the spectrum of all transmission
eigenvalues does determine spherically stratified media uniquely under mild regularity assumptions, see [1].
Since the underlying transmission problem is a non-selfadjoint eigenproblem, it may also admit complex-valued
eigenvalues in general. While numerical studies indicate their existence for quite general scatterers, see [2], theoreticaljustification has only been provided for the spherically stratified case so far, see [1]. In this talk we want to retackle thissurvey for homogeneous media and present new structural insights for complex-valued transmission eigenvalues of adisc/ball. Numerical experiments show that the concept may be extendible to other scattering shapes as well.
References
1. F. Cakoni, D. Colton and H. Haddar, Transmission Eigenvalues, Notices Amer. Math. Soc., 68, 1499-1510 (2021).
2. L. Pieronek, The method of fundamental solutions for computing interior transmission eigenvalues, Diss. BTU
Cottbus-Senftenberg (2020).

86

MINISYMPOSIUM
M8: Inverse Problems in Science and Engineering
Organizers:
Karel Van Bockstal, Ghent University, Belgium, karel.vanbockstal@ugent.be
Liviu Marin, University of Bucharest, Romania, liviu.marin@fmi.unibuc.ro
Cristiana Sebu, University of Malta, Malta, cristiana.sebu@um.edu.mt
Inverse problems arise in many areas of mathematical physics and applications are rapidly
expanding to geophysics, chemistry, medicine and engineering. This minisymposium focuses on bothanalytical and computational methods for inverse problems in Science and Engineering. The
approaches developed for such problems generally include numerical approximations, stability
analysis, proofs of uniqueness and/or existence of the solution.
The minisymposium aims at bringing together well established scientists as well as young
researchers working on inverse problems for partial differential equations. The topics of theminisymposium range from the mathematical modelling and the theoretical analysis of inverse
problems for partial differential equations where some parameters (right-hand side, kernel, diffusion
coefficient, etc.), unknown boundary condition(s) or portion of the boundary are to be found, to thedevelopment of efficient numerical schemes and their practical implementations.

87

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON 3D PET IN SPHERICAL GEOMETRY OF DATA ACQUISITION
Anwesa Deya , Gaik Ambartsoumianb , Souvik Royb , Rohit Kumar Mishraca
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA,b
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76016, USA,c
Department of Mathematics, Indian Institute of Technology-Gandhinagar, Gujarat 382055, India.

Abstract
The task of image reconstruction in positron emission tomography (PET) is mathematically equivalent
to inversion of the X-ray transform from severely undersampled and truncated data. Several methods of dealing
with these incomplete data problems have been developed throughout the years, including reprojection
algorithms and various exact and approximate rebinning techniques. However, most of these methods have beencreated and optimized for the classical cylindrical geometry of data acquisition.
This talk is dedicated to the problem of image reconstruction in PET with a spherical geometry of dataacquisition. We will discuss the major differences between the two geometries in relation to the existing
methods of image reconstruction, present some new approaches to the solution of the problem and demonstrate
the results of numerical studies.
Research reported in this presentation was partially supported by NIBIB/NINDS of the National
Institutes of Health under award number U01-EB029826. The content is solely the responsibility of the authorsand does not necessarily represent the official views of the National Institutes of Health.

88The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEM FOR QUANTIFICATION OF LOCALIZED DAMAGE USING
1D NONLINEAR ELASTIC TWO-WAVE MIXING
Pravinkumar Ghodake and Salil Kulkarni
Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra, India,
mech7pkumar@gmail.com; salil.kulkarni@iitb.ac.in

Abstract
Localized damage in a one-dimensional rod of length L is modeled as a quadratically nonlinear materialof thickness a sandwiched between two identical linear materials. Approximate analytical solutions obtained bysolving a one-dimensional nonlinear elastic wave equation indicate that the interaction of a single elastic wave
with the nonlinear material generates higher harmonics. The damage level can be measured using the higher
harmonic generation technique, but the quantification of the localized damage size has not been reported.
Approximate analytical solution for 1D backscatter and transmitted waves from quadratically nonlinear
material sandwiched between two linear materials has been obtained by Wang and Achenbach [1]. Considering
the amplitudes of the higher harmonics of the backscattered and the transmitted waves an inverse problem isformulated to estimate damage size. The inverse problem is solved by reformulating it as a constrained
optimization problem as follows:









sin 2a1c 1
sin 2a 2 c 1;
Constraints:
Minimize R21synth , a  0 (1)R
22 syntha2a1c 1
2a 2 c 1
where 1 ,  2 are angular frequencies and c is wave speed respectively.
Synthetic data ( R22 synth , R21synth ) are generated for known damage sizes and is used to estimate theactual damage sizes by using the proposed formulation. It has been observed that this approach is limited up to acertain level of damage size (e.g. 0.1-1 mm damage size in aluminum at input frequencies of 1 and 1.5 MHz). To
increase the range of the estimated damage size, the original approach has been improved by using the one way
two-wave mixing method. Approximate analytical solution for the backscattered and transmitted waves due to
two-wave mixing from quadratically nonlinear region is obtained and used to formulate a more robust inverse
problem as follows:









sin 2a 2 c 1
Minimize R22 synth ;a2a 2 c 1



sin 2a1   2 c 1
sin 2a1c 1
 R212 synth , , R21synth ,Constraints:
2a1   2 c 1
2a1c 1



(2)



sin 2a1   2 c 1
 R221 synth , , a  0 .
2a1   2 c 1
It can be observed that the additionally generated harmonics help to solve the inverse problem. Both the
optimization problems (Equation (1) and Equation (2)) are solved numerically using gradient and non-gradient
based algorithms to obtain the damage size. Using the improved approach, the detected damage size increases
from 0.1-1.0 mm to 0.1-10 mm for the same range of input frequencies. In addition, only one mixing experimentis required in the second approach as compared to two separate experiments which need to be performed for theoriginal approach.

References
1. Y. Wang and J. D. Achenbach, Reflection of ultrasound from a region of cubic material nonlinearity due toharmonic generation. Acta Mech. 229, 763–778 (2018).

89

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

NUMERICAL SOLUTION OF DIRECT AND INVERSE PROBLEMS WITH
PARAMETER UNCERTAINTY OF PHASE CHANGE ENERGY STORAGE WALL
Xingcong Dong, Haitian Yang, Yiqian He
State Key Lab of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian
University of Technology, Dalian 116024, P.R. China, heyiqian@dlut.edu.cn

Abstract
In this paper, a numerical method based on quadtree scaled boundary element method and interval
method is proposed to solve the forward and inverse heat transfer problems with uncertain parameters of phasechange energy storage wall. The quadtree scaled boundary element method is used to analyze the phase change
heat transfer directly based on the wall image, and the numerical model of the uncertain direct and inverse
problems of phase change heat transfer is established by using the interval method. In the uncertainty analysis,
the calculation of interval analysis can be significantly reduced by constructing Legendre polynomial surrogate
model to approximately replace the calculation of deterministic problems. Numerical examples show that the
proposed method can not only effectively evaluate the influence of the interval uncertainty of phase change wallmaterial parameters on the temperature change, but also obtain the material design parameters through theprescribed temperature interval.
This work provides an effective numerical model for the analysis and design of heat transfer
performance of phase change energy storage wall.

References
1. Ooi E T, Song C, Tin-Loi F, Yang Z J. Polygon scaled boundary finite elements for crack propagation
modelling, International Journal for Numerical Methods in Engineering, 2012, 91(3): 319-342

90

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

IMAGE RECONSTRUCTION IN DIFFUSE OPTICAL TOMOGRAPHY: AN
OPTIMAL BAYESIAN ESTIMATOR FOR ABSORPTION COEFFICIENT

Taufiquar Khan
University of North Carolina at Charlotte, Charlotte, NC 28223, USA,taufiquar.khan@uncc.edu

Abstract
In this talk, we will discuss an optimal Bayesian estimator including the rate of convergence toreconstruct the absorption coefficient for the Diffuse Optical Tomography (DOT). Mathematically, thereconstruction of the internal absorption or scattering coefficients is a severely ill-posed inverse problem andyields a poor quality image reconstruction.
We will present the efficacy of the proposed approach using simulations.

91

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AN ITERATIVE METHOD FOR THE CAUCHY PROBLEM FOR THE HELMHOLTZ
EQUATION

Lydie Mpinganzima
Department of Mathematics, School of Science, College of Science and Technology, University of Rwanda, P.O
Box 3900 Kigali, Rwanda, lydiem421@gmail.com

Abstract
We consider the problem of reconstructing the acoustic or electromagnetic field from inexact data given
only on an open part of the boundary of a given domain. The boundary is divided into two nonoverlapping parts.
The model problem under study is the Cauchy problem for the Helmholtz equation and it is ill-posed in the senseof Hadamard. For the reconstruction, we first introduce an artificial auxiliary boundary in the interior of thedomain and a positive parameter which are chosen to guarantee the positivity of a quadratic form associated to
the Helmholtz operator, the introduced auxiliary boundary, and the chosen parameter as it has been discussed in[1].
We present an iterative method for which two well-posed boundary value problems are solved in every
step. The algorithm works as follows: We initially choose Neumann data on an unknown part of the boundary
and Robin-type jump conditions on the interior artificial boundary. For the first problem, we solve the boundary
value problem for the Helmholtz equation where the specified Dirichlet boundary data together with the initiallychosen Neumann data on the unknown part of the boundary and the Robin-type jump conditions on the artificial
interior boundary are used. For the second problem, the boundary value problem for the Helmholtz equation issolved where the specified Neumann data is updated with Neumann data from the first solved problem andhomogeneous Dirichlet data are considered on the other part of the boundary and on the interior artificial
domain. We update the initially chosen Neumann data on the unknown part of the boundary with Neumann datafrom the previously solved problem. We also updated Robin-type jump conditions with similar conditions fromthe previously. We then solve the two problems described above.
The convergence of the proposed method will be provided. The numerical results performed using thefinite difference method will be provided. Those results show that the convergence of the proposed iterative
method require few number of iterations than the iterative algorithm presented in [1].

References
1. F. Berntsson, V.A. Kozlov, L. Mpinganzima and B.O. Turesson, An alternating iterative algorithm for the
Cauchy problem for the Helmholtz equation, Inverse Probl. Sci. Eng., 22, 45-62, 2014.

92

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STATISTICAL LINEAR INVERSE PROBLEMS BASED ON DISCRETELYSAMPLED FUNCTIONAL DATA

Mihaela Pricop-Jeckstadt
Department of Applied Mathematics, Polytechnique University of Bucharest, Romania
E-Mail: Mihaela.Pricop@upb.ro

Abstract
In this talk we investigate the performance of the spectral regularization methods when data
is a collection of discretely sampled realizations of a process modelled as a classical statistical inverse problem.
Minimax rates of convergence are established under both the common and the independent design. Different
phase transition phenomena are studied leading to a partition of the functional data into three types (non-dense,
dense and ultra-dense) in this framework.
Finally, the effect of the sampling frequency and the number of curves is illustrated in a numericalapplication.

References
1. P. Mathe, S.V. Pereverzev, Optimal Discretization of Inverse Problems in Hilbert Scales. Regularization and
Self-Regularization of Projection Methods, SIAM Journal on Numerical Analysis, Vol. 38, 2001.
2. N. Bissantz, T. Hohage, A. Munk, F. Ruymgaart, Convergence Rates of General Regularization Methods for
Statistical Inverse Problems and Applications, SIAM Journal on Numerical Analysis, Vol. 45, 2007.

93

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECONSTRUCTING MOLECULAR FLEXIBILITY
IN CRYOGENIC ELECTRON MICROSCOPY
Johannes Schwab
MRC-Laboratory of Molecular Biology, Francis Crick Avenue, Cambridge Biomedical Campus, CB2 0QH,
Cambridge, United Kingdom, schwab@mrc-lmb.cam.ac.uk

Abstract
Cryogenic electron microscopy (cryo-EM) is a powerful method to obtain the 3D structure ofmacromolecules from thousands of noisy projection images. Since these projection images emerge from manydifferent observations of the same molecule, flexible areas of it result in a local drop of resolution in a single 3dreconstruction. We propose a method based on an variational autoencoder (VAE) that uses gaussian basis
functions as a model for the molecule and estimate smooth deformation fields for every projection image. We
further use the estimated deformations to better resolve the flexible regions in the reconstruction using a filteredbackprojection algorithm along curved lines.
We present results on real data showing that we obtain improved 3D reconstruction.

94

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REAL-TIME ELECTRICAL IMPEDANCE IMAGING AT HIGH AC FREQUENCIES

Cristiana Sebu
Department of Mathematics, University of Malta, Malta
cristiana.sebu@um.edu.mt

Abstract
We investigate a multifrequency impedance imaging technique that has recently been suggested for
mammography screenings. The technique uses boundary voltages generated by time harmonic (AC) boundary
currents with either the same spatial distribution but different driving frequencies or different spatial
distributions but same driving frequency. The proposed algorithms for the complex conductivity reconstructions
are non-iterative and suitable for real-time imaging.
Numerical inversions obtained for two dimensional domains from simulated noisy data are presented.
This is joint work with Jeremy Curmi.

95

MINISYMPOSIUM
M9: Recent Advances in Analytical and Numerical Methods in Inverse
Problems for Partial Differential Equations
Minisymposium dedicated to the 70th anniversary of Professor Michael V. Klibanov
Organizers:
Christian Clason, Universität Duisburg-Essen, Germany, christian.clason@uni-due.de
Thanh Trung Nguyen, Rowan State University, USA, nguyent@rowan.edu
Paul Sacks, Iowa State University, USA, psacks@iastate.edu
Many inverse problems of practical interest involve estimating unknown parameters frommeasurements of a state described by a partial differential equation (PDE) depending on theseparameters; we mention just seismic inversion and acoustic medical imaging (which require
determining the speed of sound in a wave equation) and nondestructive testing for corrosion (where an
unknown coefficient in a Robin boundary condition has to be reconstructed). Such problems arechallenging as they are usually nonlinear and require efficient large-scale methods for their numericalsolution. In addition, analytical results are often crucial in developing appropriate numericalalgorithms, since the function space structure of PDEs is an essential part of the problem. In thisminisymposium talks by experts will be presented on recent developments in this area, including inparticular those concerning global uniqueness and global convergence methods pioneered by Professor
Klibanov.

96

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

BANG-BANG OPTIMAL CONTROL IN SPIN DYNAMICS OF RADICAL PAIRS IN
QUANTUM BIOLOGY

Ugur G. Abdullaa and Carlos Martinob,aaDepartment of Mathematical Sciences, Florida Institute of Technology, 150 West University Blvd., Melbourne,
Florida 32901, USA, abdulla@fit.edubJohns Hopkins University, Applied Physics Lab, 11100 Johns Hopkins Road, Maryland 20723, USA,
Carlos.Martino@jhuapl.edu

Abstract
Optimal control of the external electromagnetic fields and internal hyperfine parameters for the
maximization of the quantum singlet-triplet yield of the radical pairs in biochemical reactions are analyzed. The
system is modeled by Schrödinger equation with spin Hamiltonians given by the sum of Zeeman interaction andhyperfine coupling interaction terms. First- and second-order Fréchet differentiability in Hilbert spaces is proved
and the formula for the first- and second-order Fréchet derivatives are derived. Pontryagin's maximum principle
is proved and the band-bang structure of the optimal control is established. A closed optimality system for theidentification of the bang-bang optimal control is revealed. It consists of system of nonlinear ODEs for finding
the optimal state vector and its adjoint, and with subsequent identification of the bang-bang optimal control via
Pontryagin's maximum principle. We hypothesize that the bang-bang optimal control and the corresponding
optimal state vector is the quantum control theory counterpart of the quantum coherence concept, i.e. the abilityof the system to exist in several distinct states simultaneously, such as the inter-system crossing between SingletTriplet states. Several numerical algorithms are implemented for solving nonlinear optimality system, andnumerical results are demonstrated.
The results contribute towards understanding the structure-function relationship of a putative
magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic
function to inspire novel quantum devices.

97

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

CARLEMAN BASED RECONSTRUCTION ALGORITHM

Maya de Buhan
CNRS-Université Paris Saclay, France
maya.de-buhan@universite-paris-saclay.fr

Abstract
With Lucie Baudouin and Sylvain Ervedoza, we are interested in recovering coefficients inevolutionary partial differential equations. If uniqueness and stability results are generally well known, we haverecently proposed the C-bRec (for Carleman based Reconstruction) algorithm to solve these inverse problems.
Following an idea developed by Michael Klibanov, we use a Carleman estimate, a theoretical tool previously
intended to prove the well-posedness of the inverse problem, in the construction of the numerical method. In
particular, we show that the method is globally convergent, i.e. it converges to the coefficient to be recovered
whatever the initial data, thus overcoming the drawbacks of least squares methods. The C-bRec method was
initially presented in [1, 2] in the case of recovering a potential in the wave equation and generalized to therecovery of a source term in a reaction-diffusion equation ([3] with Muriel Boulakia and Erica Schwindt). In thistalk, we present in detail the case of the recovery of the propagation speed in a wave equation ([4] with Axel
Osses) from the measurement of the normal derivative of the solution on a part of the boundary.
We explain the challenges related to the numerical implementation of the algorithm and illustrate its
efficiency on one and two dimensional examples.

References
1. L. Baudouin, M. de Buhan, S. Ervedoza, Global Carleman estimates for waves and applications,
Communications in Partial Differential Equations, 38:5, pp. 823-859, 2013.
2. L. Baudouin, M. de Buhan, S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery
of a potential in the wave equation, SIAM J. Numerical Analysis, 55:4, pp. 1578-1613, 2017.
3. M. Boulakia, M. de Buhan, E. Schwindt, Recovery of a source term in the bistable reaction-diffusion equation,
ESAIM: COCV, 27 S27, 34p, 2021.
4. L. Baudouin, M. de Buhan, A. Osses, S. Ervedoza, Carleman based Reconstruction algorithm for waves,
SIAM J. Numerical Analysis, 59:2, pp. 998-1039, 2021.

98

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MULTIBAND REGULARIZATION OF A COEFFICIENT INVERSE PROBLEMFOR
THE WAVE EQUATION
Christian Clason
Universität Duisburg-Essen, Germany
christian.clason@uni-due.de

Abstract
Our results demonstrate, that learned parts in the reconstruction procedure can increase image This talkis concerned with reconstructing the spatially variable wave speed in a scalar wave equation whose values are
assumed to be taken pointwise from a known finite set. Such a property can be included in Tikhonov
regularization by a so-called “multibang” regularization term that is nonsmooth but convex.
We discuss well-posedness as well as the numerical solution of the regularized problem.

99

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOLUTION OF A THREE-DIMENSIONAL INVERSE ELASTOGRAPHY PROBLEM
FOR PARAMETRIC CLASSES OF INCLUSIONS

A. Leonova, A. Sharovb, A. YagolabaDepartment of Mathematics, National Research Nuclear University MEPhI, Moscow 115409, Russia
asleonov@mephi.ru;bDepartment of Mathematics, Faculty of Physics, Moscow State University, Moscow 119991, Russia
scharov.aleksandr@physics.msu.ru , yagola@physics.msu.ru;

Abstract
Elastography is a combination of methods of oncological examination of tissues in medicine, based on
differences in the elastic properties of healthy and tumor tissues. For example, it is well known that the Young's
modulus of tumor tissue is several times larger than the Young's modulus of healthy tissue. Usually, any of themethods of elastography can be divided into 3 stages of the study: (i) compression of tissues by external sources;(ii) measurement or calculating the displacement field of the investigated biological tissue; (iii) the calculation of
the elastic properties of the tissue by solving the inverse problem.
In this talk, we consider a quasistatic approach to elastography, in which a three-dimensional inverse problem
is solved for an isotropic linearly elastic body under small surface compressions. In this case, we have a systemof stationary partial differential equations, describing the relationship between tissue displacements and elastic
properties of the tissue, that is Young's modulus and Poisson’s ratio. To solve the problem, a special algorithm is
proposed and justified.

References
1. A.S. Leonov, A.N. Sharov and A.G. Yagola, Solution of the inverse elastography problem for parametricclasses of inclusions with a posteriori error estimate, Journal of Inverse and Ill-Posed Problems, 26, 493-500(2018).
2. A.S. Leonov, A.N. Sharov and A.G. Yagola, Solution of the inverse elastography problem in three dimensions
for a parametric class with a posteriori error estimation, Moscow University Physics Bulletin, 74, 471-480(2019).

100

MINISYMPOSIUM
M10: Computational Methods for Inverse Problems and Applications
Minisymposium dedicated to the 75th anniversary of a distinguished expert in the field of
inverse problems Professor Anatoly Yagola
Organizers:
Sergey Kabanikhin, Institute of Computational Mathematics and Mathematical Geophysics SB RAS,
Novosibirsk, Russia, ksi52@mail.ru
Maxim Shishlenin, Institute of Computational Mathematics and Mathematical Geophysics SB RAS,
Novosibirsk, Russia, maxim.shishlenin@sscc.ru
Yanfei Wang, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China,
yfwang@mail.iggcas.ac.cn
In the last decades, techniques and methods of regularization have been enriched by new
approaches including parallel computations and evolutionary algorithms. We want to bring together
researchers who are investigating fundamental approaches and modern technique in different
applications of inverse problems. In this minisymposium talks by experts will be presented on recent
developments in this area, including in particular methods pioneered by Professor Yagola.

101

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

OPTIMIZATION APPROACH IN INVERSE PROBLEMS OF DESIGNING THERMAL
CLOAKING DEVICES

Gennady Alekseeva,baInstitute of Applied Mathematics FEB RAS, 690041, 7, Radio St., Vladivostok, Russia;b
Far Eastern Federal University, 690090, 8, Sukhanova St., Vladivostok, Russia, alekseev@iam.dvo.ru;

Abstract
Transformation optics approach proposed in [1] in 2006 has been widely applied when solving problems ofmanipulation of electromagnetic and acoustic fields. Then this method has been extended to the thermal,
magnetic, electric and other fields [2, 3]. This approach allowed to design metamaterial devices for manipulatingheat flows such as invisibility cloaks, illusion devices, invertors, concentrators, etc. [see 4].
It should be noted that the technical implementation of the solutions obtained in the cited articles is
associated with significant difficulties. One way to overcome these shortcomings is to replace the original exact
cloaking problem with the problem of approximate cloaking and apply the optimization method to solve inverse
problems. Just the optimization method is used in this paper for solving inverse problems of designing thermal
devices.
The paper consists of two parts. The first part contains theoretical analysis which includes the study ofsolvability of direct and inverse problems for the heat conduction model under consideration, the derivation of an
optimality system that describes the necessary conditions for the extremum, and the establishment of important
properties of optimal solutions that are used to develop efficient numerical algorithms. Based on analysis of
optimality system we prove a number of new theorems concerning local uniqueness and stability of optimal
solutions.
The second part includes describing efficient numerical algorithms of solving problems of designing heatcloaking and shielding devices and analysis of results of numerical experiments. One can read about someobtained results in [5-7].
This work was supported by the state assignment of Institute of Applied Mathematics FEB RAS (Theme No.
075-00400-19-01).

References
1. J.B. Pendry, D. Schurig, and D.R. Smith, Controlling electromagnetic fields, Science, 312, 1780-1782 (2006).
2. F. Yang, Z.L. Mei, T.Z. Jin et al., DC electric invisibility cloak, Physics Review Letters, 109, 053902 (2012).
3. T. Han, H. Ye, Y. Luo et al., Manipulating dc currents with bilayer bulk natural matterials, Advanced
Materials, 26, 3478-3483 (2014).
4. G.V. Alekseev, Invisibility problems in acoustics, optics and heat transfer, Vladivostok, Dal’nauka, 2016.
5. G.V. Alekseev, Analysis of a two-dimensional thermal cloaking problem on the basis of optimization,
Comput. Math. Math. Phys., 58, 478-492 (2018).
6. G.V. Alekseev, D.A. Tereshko, Particle swarm optimization-based algorithms for solving inverse problems ofdesigning thermal cloaking and shielding devices, Int. J. Heat Mass Transfer, 135., 1269-1277, (2019).
7. G.V. Alekseev, V.A. Levin, D.A. Tereshko, Simulation and optimization in the problems of design ofspherical layered thermal shells, Journal of Applied Mechanics and Technical Physics, 60, N 2, 323-331,(2019).

102

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

COLLOCATION-VARIATION APPROACHES TO NUMERICAL SOLUTION OF
DIFFERENTIAL-ALGEBRAIC AND VOLTERRA EQUATIONS OF THE FIRST KIND

M. Bulatova and L. SolovarovaaaInstitute of System Dynamics and Control Theory, SB RAS, Lermontov st., 134, 664033 Irkutsk, Russia,
mvbul@icc.ru, soleilu@mail.ru

Abstract
The report deals with numerical solution of quite complicated ill-posed problems: initial problem in
differential-algebraic (DAEs) and Volterra equations of the first kind. A notion of index [1] and degree of illposedness [2], [3] characterize the complexity of the DAEs and Volterra equations, respectively.
It is known that many methods developed for ordinary differential equations and Volterra equations of thesecond kind can be unstable or inapplicable for the problems under consideration.
We propose new collocation-variation approach to numerical solution of these classes of equations andformulate conditions and rate of convergence. We investigate these methods for A- and L- stability.
This work was partially supported by the Russian Foundation for Basic Research, projects 18-29-10019-mk,
18-01-00643-a.

References
1. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic
Problems, Springer-Verlag, Berlin, 1996.
2. A. Apartsyn, Nonclassical Linear Volterra Equations of the First Kind, VSP, Utrecht, 2003.
3. A. Apartsyn and A. Bakushinskii, Approximate solution of Volterra integral equations of the first kind by thequadrature method, Differ. Integral Eqns., 1, 248–258 (1972) (in Russian).

103

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

UNIFORM STABILITY OF RECOVERING THE DIRAC OPERATOR WITH AN
INTEGRAL DELAY FROM THE SPECTRUM

S.A. Buterin
Department of Mathematics, Saratov State University, Saratov, Russia
buterinsa@info.sgu.ru

AbstractLetbe the spectrum of the boundary value problemof the form
(1)

where

is the spectral parameter andwhile the functions
are complex-valued, and.
Inverse Problem 1. Given the spectrum, find the functions
and.
In [1], the unique solvability of Inverse Problem 1 was established and its constructive solution was obtained.
In particular, we proved that, for any complex sequence
to be the spectrum of a boundary value problemof the form (1), it is necessary and sufficient to have the asymptotics
In this note, we establish uniform stability of Inverse Problem 1, i.e. the following theorem holds.
Theorem 1. For any, there exists
such thatas soon as

and, whereis the spectrum of.

Let us note that uniform stability of inverse problems for classical Sturm–Liouville and Dirac operators wasstudied in [2–4]. Here, we use a different approach relying on the uniform stability of the so-called main
nonlinear equation of the inverse problem [5] as well as on the uniform stability of recovering the characteristic
determinant of the problem
from its zeros [6]. Finally, note that uniform stability of inverse problems forsome scalar integro-differential operators was studied in [7, 8].
This research was supported by Russian Science Foundation, Grant No. 22-21-00509.

References
1. N.P. Bondarenko, S.A. Buterin, On recovering the Dirac operator with an integral delay from the spectrum,
Results in Mathematics, 71 (3), 1521–1529 (2017).
2. A.M. Savchuk, A.A. Shkalikov, Inverse problems for Sturm–Liouville operators with potentials in Sobolev
spaces: uniform stability, Funk. Anal. Appl. 44 (4), 270–285 (2010).
3. R.O. Hryniv, Analyticity and uniform stability in the inverse singular Sturm–Liouville spectral problem,
Inverse Problems 27, art. no. 065011 (2011).
4. R.O. Hryniv, Analyticity and uniform stability in the inverse spectral problem for Dirac operators, J. Math.
Phys. 52, art. no. 063513 (2011).
5. S. Buterin, M. Malyugina, On global solvability and uniform stability of one nonlinear integral equation,
Results in Mathematics, 73, art. no. 117 (2018).
6. S.A. Buterin, On the uniform stability of recovering sine-type functions with asymptotically separated zeros,
Math. Notes 111 (3), 343–355 (2022).
7. S. Buterin, Uniform stability of the inverse spectral problem for a convolution integro-differential operator,
Appl. Math. Comput. 390, art. no. 125592 (2021).
8. S. Buterin, Uniform full stability of recovering convolutional perturbation of the Sturm–Liouville operator
from the spectrum, J. Diff. Eqns. 282, 67–103 (2021).

104

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEM CHALLENGES IN PRACTICAL SUBSURFACEGEOENGINEERING APPLICATIONS

Vasily Demyanov,
Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh
v.demyanov@hw.ac.uk

Abstract
The problem of subsurface reservoir prediction is naturally inverse as it is base inferring unknown subsurface
properties given the dynamic Earth response and geological prior information. Reservoir predictions are madeusing porous media flow models calibrated to past observations, referred to as to history matching. Several
challenges associated with implementing practical history matching solutions for decision making are: i) vast
subsurface uncertainty not limited only to non-Gaussian priors of the unknown porous media characteristics(porosity, permeability, stratigraphic features, structural fault properties) but also to the mathematical descriptionof the spatial porous properties’ distributions; ii) complex non-Gaussian behavior of the likelihood function,
which in practical decision making has a multivariate nature; iii) difficulty in decoupling numerical errors,parameter uncertainty, model inadequacy and data discrepancy to make inference prediction; iv) accurate
posterior inference given large computational cost of full physics modelling and the limitation of the surrogate
proxies.
The presentation will go over real examples that demonstrate the importance of tackling the outlined
challenge for practical decision making in subsurface resources development. In particular, the presentation willreflect up on:i)
the importance of informative priors to ensure geological realism of inverse modelling solutions;ii) ii) the value of multi-objective optimization and its approximate posterior inference;iii) iii) the impact of the inverse model solution quality in terms of likelihood and accurate estimation of
the posterior for optimal development decisions.

References
1. D Arnold, V Demyanov, T Rojas, M Christie (2019) Uncertainty Quantification in Reservoir Prediction: Part
1—Model Realism in History Matching Using Geological Prior Definitions, Mathematical Geosciences 51 (2),209-240
2. V Demyanov, D Arnold, T Rojas, M Christie (2019) Uncertainty quantification in reservoir prediction: Part
2—Handling uncertainty in the geological scenario, Mathematical Geosciences 51 (2), 241-264
3. B Duarte, V Demyanov, L Pereira (2019) The Value of History Matching in Field Development, 81st EAGE
Conference and Exhibition, London, 2019.
4. J Hutahaean, V Demyanov, M Christie (2019) Reservoir development optimization under uncertainty for infill
well placement in brownfield redevelopment, Journal of Petroleum Science and Engineering 175, 444-464

105

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON ACCELERATION OF LANDWEBER METHOD FOR ILL-POSED PROBLEMS
U. Hämarik, U. Kangro, T. Raus
Institute of Mathematics and Statistics, University of Tartu, Estonia
uno.hamarik@ut.ee, urve.kangro@ut.ee, toomas.raus@ut.ee
Abstract
We consider linear illposed problem

f , A L( H , F ), f R ( A),
where H , F are Hilbert spaces and range R ( A) is non­closed, kernel N ( A) is non­trivial. Only noisyAudata ff

are available instead of exact data f , f

 . The approximate solution may be found by

the Landweber iteration method

un 1

un

A ( f Aun ), 

1

(0,

A

2

), n

N0 .

(1)

In this method and in many other analogous methods iterations are often stopped via discrepancy principle on
first indexnn( ) , for which Aunb , bf

1 . Then method (1) has order optimal error

estimate on the source­like solutions, but often needs huge number of iterations.
Articles [1,2] propose and analyze the Nesterov acceleration for Landweber methodzn  A ( f 
In [2]  1 is fixed.
un 1

Azn ), znxn n (unun 1 ),  n(n 1) / (n ), 

1. (2)

Stopping iterations via the discrepancy principle, this method needs about square root of number of
iterations, needed in method (1). Papers [1, 2] show that fixed 
const in (2) leads to saturation effect inconvergence rate.
We propose some alternative choices of  n depending not only on k , but also on residual. One example
is

n

n
(1  n 1 ),  n
n 1

min max

n 1

n
, n1
,1 ,  n
1 n 1 2n 1 n

Aun

f .

Extensive numerical experiments show that for proposed choices of  n the accuracy of the approximation issimilar to the method (1) and number of iterations is similar to the method (2).

References
1. A. Neubauer, On Nesterov acceleration for Landweber iteration of linear ill­posed problems, J. Inverse
Ill­Posed Probl., 25, 381–390 (2017).
2. S. Kindermann, Optimal­order convergence of Nesterov acceleration for linear ill­posed problems, Inverse
Problems, 37, 065002 (2021).

106

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

CARLEMAN ESTIMATES FOR GLOBALLY CONVERGENT NUMERICALMETHODS FOR COEFFICIENT INVERSE PROBLEMS

Mikhail Victor Klibanov,
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223,
USA, mklibanv@uncc.edu

Abstract
Since the field of Inverse Problems is an applied one, it is insufficient just to prove some theorems.
Rather it is necessary to develop reliable numerical methods. However, conventional numerical methods for
Coefficient Inverse Problems (CIPs) are unreliable. The reason is that they are based on the minimization of leastsquares cost functionals. These functionals are nonconvex. Therefore, as a rule, they have many local minima
and ravines. Since any minimization procedure can stop at any local minimum, which can be far from the true
solution, then these methods are unreliable and unstable.
In the past several years Klibanov and his research team have successfully developed a radically new
and very effective numerical method of solving CIPs. Furthermore, this method is verified on a variety of
microwave experimental data. This is the so-called "convexification" method. In the convexification one
constructs a globally strictly convex weighted Tikhonov-like functional. Therefore, the problem of local minima
is avoided. Furthermore, the convexification generates globally convergent numerical methods. The key to thisfunctional is the presence in it of the so-called Carleman Weight Function. This is the function which is involved
as the weight in the Carleman estimate for the corresponding Partial Differential Operator.
The convexification will be presented for a broad variety of CIPs. Numerical results will also be presented forboth computationally simulated and experimental data

107

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON DISCREPANCY PRINCIPLES FOR TIKHONOV REGULARISATION
Marek A. Kojdecki
Institute of Mathematics and Cryptology, Military University of Technology, 00-908 Warsaw, Poland,
Marek.Kojdecki@wat.edu.pl; m_kojdecki@poczta.onet.pl;

Abstract
The linear operator equation of the first kind:with a bounded linear operator, between
Hilbert spaces and , to which a generalised solution
exists, when only an equation:with inexact
data(a bounded linear operator,, together with an estimateof error indata:,) is known, is the abstract form for linear ill-posed problems. It can beeffectively solved by Tikhonov's regularisation method as a variational problem of seeking the unique vector
minimising the smoothing functional:
with the regularisation parameter[1-6],
that can be found as the solution of the linear operator equation of the second kind:.
(1)
Particularly, the inexact data can be finite-dimensional approximations to, additionally perturbed as aresult of preceding measurements or computations; the estimatesshould then include all-type errors indata. A question of both theoretical and practical importance is the choice ofenabling the convergence. Effective rules for choosing the regularisation parameter are usually defined in the form of
additional equation (2) to be satisfied by pair(together with (1)). The most usefull are the discrepancy
principle [1,2,3], especially the generalised discrepancy principle [2]:; (2a)or the second discrepancy principle [4]:.
(2b)Another rule can be formulated as a combination of the Tikhonov regularisation and the method of quasisolutions [5].
All regularisation algorithms with a-posteriori parameter choice rules include iterative processes forsolving non-linear equations like (2a) or (2b) together with (1), combined with discretisation of equations (1) and(2). These algorithms are different to those of iterative regularisation constructed immediately for equation (1)[6]. The properties of methods with discrepancy principles and especially some new results for seconddiscrepancy principle will be considered and commented in the talk.

References
1. V.A. Morozov, Methods for solving incorrectly posed problems, New York – Berlin – Heidelberg, Springer,
1984.
2. A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov and A.G. Yagola, Numerical methods for the solution of illposed problems, Kluwer Academic Publisher, 1995.
3. A.B. Bakushinsky, A.V. Goncharsky, Ill-posed problems: theory and applications, Dordrecht – Boston –London, Kluwer, 1994.
4. M.A. Kojdecki, New criterion of regularisation parameter choice in Tikhonov's method, Biuletyn WAT (Biul.
Mil. Univ. Technol.), XLIX (1), 47-126 (2000).
5. V.K. Ivanov, V.V. Vasin, V.P. Tanana, Theory of linear ill-posed problems and its applications, Utrecht,
VSP, 2002.
6. G.M. Vainikko, A.Y. Veretennikov, Iteration procedures in ill-posed problems, Moscow, Nauka, 1986, in
Russian.

108

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

HIGH-PERFORMANCE REGULARIZATION OF MULTI-PARAMETRIC INVERSE
PROBLEMS OF EPIDEMIOLOGY AND SOCIAL NETWORKS

O. Krivorotkoa,b, S. Kabanikhina,b, V. Kashtanovaa, T. Zvonarevab and D. BykovbaInstitute of Computational Mathematics and Mathematical Geophysics of SB RAS, 6 Prospect Academika
Lavrentijeva, 630090 Novosibirsk, Russian Federation, krivorotko.olya@mail.ru, kanabikhin@sscc.ru,
vikakashtanova@ya.ru;bNovosibirsk State University, 1 Pirogova street, 630090 Novosibirsk, Russian Federation,
t.zvonareva@g.nsu.ru, batyan@my.com;

Abstract
Mathematical models in natural sciences based on mass balance law and diffusive processes are described bysystems of ordinary differential equations (ODE) and partial differential equations of parabolic type (PDE).
Considered mathematical models are driven by a lot of parameters such as coefficients of ODE/PDE, initialconditions, source, etc., that play a key role in prediction properties of models. The most of parameters is
unknown or can be rough estimated. The inverse problem consists in identification of parameters ofmathematical models using additional measurements of some direct problem statements in fixed times.
Considered inverse problems are ill-posed, i.e. its solutions could be non-unique and unstable to data errors. The
identifiability and sensitivity analysis are used to construct the regularization method for solving of inverse
problems [1].
Inverse problems are formulated as minimization problems of loss function depends on vector of parameters.
To find the global minima of minimization problems the combination of machine learning (ML), heuristic(simulated annealing, particle swarm optimization, genetic algorithm, etc.) and deterministic approaches are
implemented. The ML methods such as artificial neural network, support vector machine, etc., as well asheuristic ones identify the global minima domain, could be easily paralleled and do not use the loss function
features. Then the gradient deterministic methods identify the global minima in its area with guaranteed accuracy[2, 3]. The continuous and discrete gradient of loss function are derived with its convergence estimates [4]. On
the other hand, the loss function can be represented as a multi-scale tensor to which the tensor train (TT)decomposition can be applied. TT method is easily paralleled and uses the structure of the loss function [5]. The
confidence intervals for control an accuracy of approximate inverse problem solution are constructed andanalyzed [3].
The numerical results for solving of inverse problems for mathematical models of epidemiology and social
processes in online social networks are presented and discussed.
This work is supported by the Russian Science Foundation (project No. 18-71-10044) and by the the grant of
President of Russian Federation (Agreement No. 075-15-2019-1078 (МК-814.2019.1)).

References
1. H. Miao, X. Xia, A.S. Perelson, H. We. On identifiability of nonlinear ODE models and applications in viral
dynamics, SIAM Rec. Soc. Ind. Appl. Math., 53(1), 3-39 (2011).
2. S. Kabanikhin, O. Krivorotko, V. Kashtanova. A combined numerical algorithm for reconstructing themathematical model for tuberculosis transmission with control programs, Inverse Ill-posed Probl., 26(1), 121131 (2018).
3. H. Banks, S. Kabanikhin, O. Krivorotko, D. Yermolenko, A numerical algorithm for constructing an
individual mathematical model of HIV dynamics at cellular level, Inverse Ill-posed Probl., 26(6), 859-873(2018).
4. P. Dvurechensky and A. Gasnikov, Stochastic intermediate gradient method for convex problems withstochastic inexact oracle, J. Optim. Theory Appl., 171(1), 121-145 (2016).
5. V.V. Zheltkova, D.A. Zheltkov, Z. Grossman, G.A. Bocharov, E.E. Tyrtyshnikov, Tensor based approach to
the numerical treatment of the parameter estimation problems in mathematical immunology, Inverse Ill-posed
Probl., 26(1), 51-66 (2018)

109

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARIZING METHODS FOR PREDICTING BULKY MOLECULESVIBRATIONAL SPECTRA BASED ON THE COMBINED USE OF AB INITIO AND
EXPERIMENTAL DATA
I. Kochikova, G. Kuramshinab , S. Sharapovaaa
Scientific Research Computing Centre, Lomonosov Moscow State University, Moscow 119991, Russia
igor@kochikov.ru svet.sharapova@gmail.comb
Department of Physical Chemistry, Faculty of Chemistry, Lomonosov Moscow State University, Moscow,
119991, Russia, kuramshi@phys.chem.msu.ru

Abstract
A number of different mathematical problems arise in the data processing of experimental data obtainedby means of infrared and Raman spectroscopy. The most important is the so-called inverse vibrational problem
of determining parameters of the molecular force field (force constants) from given experimental data
(vibrational frequencies, isotope frequency shifts, Coriolis constants, centrifugal distortion constants, etc.). The
accumulation of data on force constants is necessary for prediction of spectra and other properties of the bulky
compounds not yet investigated which are too large for acuurate quantum mechanical calculations.
The mathematical problem of calculating molecular force fields within the general approximation ofsmall vibrations (harmonic model) is an ill-posed problem and it does not satisfy any of the three well-posedness
conditions (existence of solution, its uniqueness and stability to perturbations in input data). In most cases, themain difficulty is non-uniqueness of solution. Different algorithms based on the theory of regularization of
nonlinear ill-posed problems have been proposed for solviing this problem and finfing the sets of force constants[1]. In our strategy the stabilizing matrix F0 is chosen as a result of quantum mechanical calculations, and thus
we search for matrix F which is the nearest by the chosen Euclidean norm to the given ab initio F0. The
optimized solution is referred to as Regularized Quantum Mechanical Force Field (RQMFF). New numericalalgorithm for the calculation of scale factors for the molecular force fields expressed in Cartesian coordinates isdeveloped and implemented as a part of the software package SPECTRUM [2,3].
The importance of large molecular systems (biological objects, polymers, giant aggregates etc.)
stimulates the development of special approaches for the describing their physicochemical properties such as
molecule geometry, vibrational frequencies, and thermodynamic functions, etc. Fast growing computational
resources and numerical methods lead to the great advantage of modern methods of quantum chemistry for thesolving many problems of structural chemistry in application to the large molecular systems. But the accurate
calculations of the large molecular systems consisting of a few hundred atoms are still limited by the dimensionsof systems because the purely ab initio methods require very large computtaional resources. One of possible
approaches in such cases is to compose the matrix of force constants of bulky molecules from separate fragments
corresponded to the smaller size units of force constants calculated as the regularized quantum mechanical force
field (RQMFF) or regularized scaled quantum mechanical force field (RSQMFF). Results obtained within thisapproach are demonstrated for some melatonin and pyridoxale derivatives.
This work was supported in part by the RFBR grant No 18-03-00412a.
References
1. A.G.Yagola, I.V.Kochikov, G.M.Kuramshina, Yu.A.Pentin. Inverse problems of Vibrational Spectroscopy.
VSP. Zeist, The Netherlands, 1999, De Gruyter, 2014.
2. I.V.Kochikov,G.M.Kuramshina,A.V.Stepanova, Int. J. Quant. Chem. 109, 28-33 (2009)..
3. I.V.Kochikov,G.M.Kuramshina,A.V.Stepanova, Struct. Chem. 30 (2), 605-614 (2019).

110

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

LOCAL SOLVABILITY AND STABILITY OF INVERSE SPECTRAL PROBLEM
FOR NON-SELF-ADJOINT STURM-LIOUVILLE OPERATOR
M. Kuznetsova
Department of Mathematics, Saratov State University, Saratov, Russia,
kuznetsovama@info.sgu.ru

Abstract
We study the inverse problem of recovering a complex-valued square-integrable potential in the Sturm–
Liouville equation from spectra of two boundary value problems
with one common boundary condition:y ( x)
y (0)
Let

nj

n 1

q ( x) y ( x)( j)y ( ),y ( x), xj(0, ),

0,1.

be the spectrum of L j (q), j

0,1 It is known that the potential is uniquely determined by these

two spectra. In the self-adjoint case (when q( x) is real-valued), the inverse problem of recovering the potential
q from two spectra is well-studied. Borg was first who studied this inverse problem [1]. He has proved its local
solvability and stability of the solution. In the self-adjoint case, there was also obtained the global solvability [2],which includes necessary and sufficient conditions on any sequences to be the spectra of some boundary
problems L j (q) . This result was generalized in [3] to the potentials from the continuous scale of Sobolev
spaces, and uniform stability has been obtained under some assumptions.
For the complex-valued potentials, the analogs of classical Borg’s results forq

L2 (0, ) wereproved in [4]. The difficulty which arises there is that the multiple values in the spectra can occur. To overcome
it, we generalised classical Borg’s method to the case of multiple spectra. Using this approach, we proved
solvability and stability of the inverse problem for complex-valued smooth potentials:

W21 (0, ) there exists

Theorem 1. For any complex-valued model potential qsequences

nj

n 1

, j

0 such that if arbitrary

0,1 satisfy the condition

n2

:

2
n0

n0

2
n1

n1

,

n 1

then there exists a unique function q
problems L j (q), j

W21 (0, ) such that0,1 correspondingly. Moreover, q

nj

n 1

q W 1 (0, )2

are the spectra of the boundary value

C , where the constant C depends

only on q .
This work was supported by Grant 20-31-70005 of the Russian Foundation for Basic Research.

References
1. G. Borg, Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Acta Math., 78, 1–96 (1946).
2. V. A. Marchenko and I. V. Ostrovskii, A characterization of the spectrum of the Hill operator. Math. USSRSb., 26, 493–554 (1975).
3. A. M. Savchuk and A. A. Shkalikov, Inverse problems for Sturm–Liouville operators with potentials in
Sobolev spaces: uniform stability, Funk. Anal. Appl., 44, 270–285 (2010).
4. S. Buterin and M. Kuznetsova, On Borg’s method for non-selfadjoint Sturm–Liouville operators, Anal. Math.
Phys., 9, 2133–2150 (2019).

111

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FAST COMPUTATION OF TSUNAMI WAVE PARAMETERS BY MEASUREDDATA INVERSION

M. Lavrentieva,b, An. Marchuka, M. Shadrina,ba
IAE SB RAS, 1 Ac.Koptyug prospect, 630090 Novosibirsk, Russia, mmlavrentiev@gmail.com
b
NSU, Pirogov st., 1, 630090 Novosibirsk, Russia, mmlavr@nsu.ru

Abstract
Speeding up calculation of tsunami wave propagation is very important for the events in a near fieldzone. This is just the case of Japan, where it takes about 20 minutes after the quake for the wave to approach thenearest coast. Use of the field programmable gates array (FPGA) microchip makes it possible to achieve
valuable performance gain with the modern regular personal computer. With the help of such a tool, we study
the distribution of tsunami wave maximal heights along the coast at Southern part of Japan depending on
particular location of tsunami source. Synthetic source covers area of 100x200 km rectangle and has realistic
shape. As is observed numerically, only selected parts of the entire coast line are subject of dangerous tsunami
wave amplitude. Particular locations of such areas strongly depend on location of tsunami source. It is possible
to provide local authorities with the PC based software/hardware tool for fast (say, in a few minutes) evaluationof tsunami danger for particular village or industrial unit at the coast [1,2].
Source parameters of tsunami waves are an essential part of any modern tsunami warning system.
Recalculation of a measured time series (wave profile obtained by a seabed based pressure sensor) in terms ofinitial sea surface displacement at tsunami source is among the most (or) one of the promising approaches to be
applied in a warning center. The “orthogonal decomposition”, that was proposed and studied earlier by theauthors, is numerically studied here. Realistic shape of sea surface displacement and digital bathymetry of thesouthern part of Japan are used. To study functionality of the proposed approach, wave profiles are obtained at
positions of the sensors of DONET – Dense Oceanfloor Network system for Earthquakes and Tsunamis –pressure gauge network of Japan. We stress on the quality of tsunami source parameters reconstruction as well as
on time required. As observed, just a part of the first wave period is enough for robust determination of such
source parameters as amplitude and total volume of water.
Results of numerical tests are summarized in table and then discussed [3,4].

References
1. M. Lavrentiev, K. Lysakov, An. Marchuk, K. Oblaukhov, M. Shadrin, FPGA-based Modelling of the Tsunami
Wave Propagation at South Japan Water Area, in Proc. OCEANS'18 MTS/IEEE Kobe, Japan, May 28-31,
2018.
2. M. Lavrentiev, A. Romanenko, K. Oblaukhov, An. Marchuk, K. Lysakov, M. Shadrin, FPGA Based Solution
for Fast Tsunami Wave Propagation Modeling, in Proc. 27th International Ocean and Polar Engineering
Conference, 25-30 June, San Francisco, California, 924-929, 2017.
3. D. Kuzakov, M. Lavrentiev, An. Marchuk, Determination of Initial Sea Surface Displacement at Tsunami
Source by a Part of Wave Profile, in Proc. OCEANS'18 MTS/IEEE Kobe, Japan, May 28-31, 2018.
4. M. Lavrentiev, D. Kuzakov, A. Romanenko, A. Vazhenin, Determination of Initial Tsunami Wave Shape at
Sea Surface, in Proc. OCEANS 2017-Aberdeen, IEEE, 1-7, 2017.

112

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ASIMPTOTIC SOLUTIONS OF INVERSE COEFFICIENT PROBLEMS FOR
BURGERS TYPE EQUATIONS WITH INTERIOR LAYERS

Nikolay N. Nefedov
Lomonosov Moscow State University, Russia
nefedov@phys.msu.ru

Abstract
The possibility of applying asymptotic analysis methods to solving inverse problems for a nonlinear
singularly perturbed Burgers-type equation (reaction-diffusion-advection equation) with time-periodic
coefficients is shown. In particular, the problems of reconstructing the linear gain (reaction coefficient)describing the properties of the medium and the boundary conditions from known information about the observed
solution of the direct problem over a certain time interval (period) are considered. Similar problems arise in gas
dynamics, in nonlinear wave theory, biophysics, chemical kinetics, and many others practical applications and aredescribed by non-linear parabolic equations with small parameters in the derivatives (see, for example, [1] - [4],and references therein). These problems are intensively studied in connection with the fact that they act asmathematical models that reveal the main mechanisms that determine the behavior and more complex models of
nonlinear wave theory.
Recent results for some classes of initial boundary value problem for some classes of Burgers-type
equations, for which we investigate moving fronts by using the developed comparison technique are presented.
These results were illustrated by the problem

 2u
u u
 A(u, x, t ) (u, x, t ), x  (0,1), t  R,
2
x
x
t
u(0, t ,  )  u 0 (t ), u(1, t ,  )  u1 (t ), t  R,u ( x, t ,  )  u ( x, t  T ,  ) x  0,1, t  R,



The work demonstrates further development asymptotic-numerical methods for solving direct andinverse problems with boundary and inner layers (see also [5], [6]. This approach is applied to a new class oftime-periodic reaction-diffusion-advection problems with inner transition layers. The concept of an asymptotic
solution of inverse coe_cient problems is introduced. This approach is demonstrated on the inverse coe_cient
problem. For this equations the problem is reduced to a much simpler problem. This allows us to classify theoriginal problem as a well-posed or ill-posed inverse problem. The proposed approach can be applied to
sufficiently wide class of problems with boundary and inner layers.
This work was supported by the Russian Science Foundation (project N18-11-00042).

References
1. N.N. Nefedov, Comparison Principle for Reaction-Diffusion-Advection Problems with Boundary and Internal
Layers, Lecture Notes in Computer Science, 8236 (2013), pp. 62 - 72.
2. N.N. Nefedov, L. Recke, K.R. Schnieder, Existence and asymptotic stability of periodic solutions with aninterior layer of reaction-advection-diffusion equations, Journal of Mathematical Analysis and Applications,
405 (2013), pp. 90 - 103.
3. N.N. Nefedov, E.I. Nikulin, Existence and stability of periodic contrast structures in the reaction-advectiondi_usion problem, Russian Journal of Mathematical Physics, 22(2) (2015), pp. 215 - 226.
4. N.N. Nefedov and O. V. Rudenko, On front motion in a Burgers-type equation with quadratic and modular
nonlinearity and nonlinear amplification, Doklady Mathematics, 97(2018), pp. 99 – 103.
5. D. V. Lukyanenko, M. A. Shishlenin, V. T. Volkov. Solving of the coefficient inverse problems for anonlinear singularly perturbed reaction-diffusion-advection equation with the final time data, Communicationsin Nonlinear Science and Numerical Simulation, 54 (2018), pp. 233-247.
6. D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, and A. G. Yagola. Application of asymptotic analysis forsolving the inverse problem of determining the coefficient of linear amplification in Burgers' equation, Moscow
University Physics Bulletin, 74(2) (2019), pp. 131-136.

113

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FINITE DIFFERENCE REGULARIZED METHOD FOR SOLVING THE
GENERALIZED ONE-DIMENSIONAL INVERSE PROBLEM OF PROPAGATION
OF THE ACTION POTENTIAL ALONG NERVOUS FIBER
A. Satybaeva, G.S. Kurmanalievaa and T.Ch. Kultaevba
Osh Technological University, 723503, Osh city, N. Isanov street 81, Kyrgyzstan, abdu-satybaev@mail.ru,
gulzat-kurmanalieva@mail.ru;b
Osh State University, 723500, Osh city, V.I. Lenin 331, Kyrgyzstan, kt_1958.2005@mail.ru;

Abstract.
We study the process of propagation of a nerve impulse along a nerve fiber in a one-dimensional case with an
instantaneous source and a flat boundary. The process is described by a parabolic equation problem and, using
the Laplace transform, is reduced to a hyperbolic equation problem. The problem of recovering the unknown
coefficients of the equation is considered.
Using the characteristic method and the singularity extraction method, the inverse hyperbolic problem isreduced to a regular inverse problem with data on characteristics. A finite-difference method was applied to thelast problem and the solution of the inverse problem was regularized.
A finite-difference regularized solution of an unknown coefficient is obtained and it is a finite-difference
regularized solution of an inverse problem of parabolic type.
The convergence estimate is obtained and the convergence of the regularized solution of the inverse problem
to the exact solution of the inverse problem is shown.
Formulation of the problem: The process of propagation of a nerve impulse along a nerve fiber in a onedimensional case is described by a telegraph equation of a parabolic form [1]:C m x r  x   2 u  x, t  u  x, t u x, t  a, x  R  , t  R ,t
2  a x  x 2
 m x   l(1)

  – is the intracellular action potential,  x ,  x – is the resistivity of the plasma(membrane) and nerve fiber, ra x  – is the radius of the nerve fiber, C x – is the capacity per unit area of thewhere u x, t

m

a

m

membrane, l – is the thickness of the membrane, a , m – are the axons (nerve fiber) and membrane indices.
The inverse parabolic problem consists in determining one of the parameters ra x  a x  m x Cm x 
with the remaining parameters known and for the given initial and boundary conditions of the form:ux, t  t 0  0 , u x x, t  x0  h0 t   r01 t   p0 2 t  , t  R ,h0 , r0 , p0 –
 2 t  positively constant numbers,(2) t  – is the Heaviside theta function, 1 t   t t  ,2t t  , as well as given additional information in the form:2ux, t  x0  g t  , t  0,2T  ,(3)where T – time, is a positively constant number. Here the / sign means or, i.e. one of the coefficients isdetermined with the remaining coefficients known.
We use the Laplace transform [2] to solve problem (1) - (3), that is,u x, t    Vt x, G t , d   V x, Gtt t , d , where G t ,  
0

0

1

t

2

e 4t is the Green's function, inverse

problem (1) - (3) is reduced to a hyperbolic inverse problem and this inverse problem was solved by the finitedifference regularized method.
References
1. A.L. Hodgkin, A.F. Huxley A quantative description of membrane current and its application to conductionand excitation in nerve fiber // J. Physiol. (London). V.117. N4. P. 500-544, (1952).
2. S.I. Kabanikhin. Inverse and ill-posed problems. Novosibirsk. Siberian Scientific Publishing House, p. 457,
2009.

114

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEMS IN THIN FILM OPTICS: RECENT ACHIEVEMENTS AND
THE NEWEST TRENDS
A.V. Tikhonravov
Research Computing Centre, Lomonosov Moscow State University, 119991 Moscow, Russian Federation,
tikh@srcc.msu.ru
Abstract
Thin film optical coatings can be found in virtually all modern scientific instruments and consumer
products. Examples range from high quality coatings for LIGO interferometer mirrors, space telescopes, nuclear
fusion facilities, to bulk coatings for mobile phones, eyeglasses, architectural glass, and more. Coating
technology has made tremendous progress over the past decades, and this is largely due to the progress insolving inverse problems in thin film optics. Two main classes of these problems are distinguished: synthesis
problems associated with the design of coatings with given spectral characteristics, and a wide class of inverse
reconstruction problems associated with various aspects of the practical production of optical coatings. Modern
coatings can have many dozens and even hundreds of layers. The thicknesses of these layers are the main design
parameters of optical coatings that determine their spectral characteristics. Due to the large number of design
parameters, designing of an optical coating is an extremely difficult inverse synthesis problem; however,modern design methods make it possible to design coatings of any type [1, 2]. Moreover, in many cases, multiple
designs can be obtained, potentially allowing the selection of the design with the best feasibility properties. The
choice of such a design is closely related to the choice of a monitoring procedure for controlling the coating layer
thicknesses during the production process. At present, various optical monitoring techniques are widely used tocontrol the production of the most complex optical coatings [3]. Optical monitoring procedures are based on
measuring the reflectance or transmittance of a coating during its deposition, either at one monitoring
wavelength (monochromatic monitoring) or at a dense wavelength grid in a certain spectral region (broadband
monitoring). The practical implementation of these procedures requires the solution of multiparametric inverse
reconstruction problems. In the last few years, a new class of algorithms, called non-local algorithms, has been
proposed to solve these problems [4, 5]. A distinctive feature of the non-local algorithms is that they utilize ahuge amount of all the experimental data accumulated during the coating deposition process. They also makeextensive use of the basic ideas of the regularization theory related to the stabilization of the solution to theinverse problem. As a result, the accuracy of solving inverse reconstruction problems has noticeably improved.
Another direction in the production of high-quality optical coatings is the choice of the optimal combination of
the theoretical coating design and the monitoring procedure used for its practical implementation. The
theoretical basis for optimizing this choice is provided by recent results on the correlation of errors caused bydifferent optical monitoring procedures and the error self-compensation effect associated with this correlation [68]. A universal approach to the study of the error correlation has been developed and an estimate for the strengthof the error self-compensation effect has been proposed. Using this estimate, the combination of design andmonitoring procedure that provides the best feasibility properties is selected.
References
1. A. Tikhonravov, Design of Optical Coatings, in Optical Interference coatings, Eds. N/ Kaiser and H.K.
Pulker, Springer- Verlag, Berlin, pp. 81-104, 2003.
2. A. Tikhonravov and M. Trubetskov, Modern design tools and a new paradigm in optical coating design,
Appl. Opt. 51, pp. 7319-7332, 2012.
3. A. Tikhonravov, M. Trubetskov, T. Amotchkina, in Optical Thin Films and Coatings, 2nd edition, 2018,
Chapter 3, Optical monitoring strategies for optical coating manufacturing, pp. 65-101.
4. I. Kochikov, Iu. Lagutin, A. Lagutina, D. Lukyanenko, A. Tikhonravov, A. Yagola, A.G, Raising theaccuracy of monitoring the optical coating deposition by application of a nonlocal algorithm of data analysis, J.
Appl. Ind. Math. 14, 330–339, 2020.
5. I. Kochikov, Iu. Lagutin, A. Lagutina, D. Lukyanenko, A. Tikhonravov, S. Sharapova, A. Yagola,
Comparative analysis of algorithms for solving inverse problems related to monochromatic monitoring thedeposition of multilayer optical coatings, Computational Mathematics and Mathematical Physics, 61(9):1504–
1510, 2021.
6. A. Tikhonravov, I. Kochikov, and A. Yagola, Error self-compensation mechanism in the optical coatingproduction with direct broad band monitoring, Opt. Express, 25, 27225-27233, 2017.
7. A. Tikhonravov, I. Kochikov, and A. Yagola, Mathematical investigation of the error self-compensation
mechanism in optical coating technology, Inverse Problems in Science and Engineering, Vol. 26. P. 1–16, 2017.
8. A. Tikhonravov, A. Lagutina, Iu. Lagutin, D. Lukyanenko, I. Kochikov, and A. Yagola, Self-compensationof errors in optical coating production with monochromatic monitoring, Opt. Express 29, 44275-44282, 2021.

115

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON THE GENERALIZED ASYMPTOTICAL REGULARIZATION FOR LINEAR ILLPOSED PROBLEMS

Ye Zhanga,baShenzhen MSU-BIT University, 518172 Shenzhen, China, ye.zhang@smbu.edu.cn;bSchool of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China.

Abstract
In this talk, two new classes of iterative regularization methods for linear ill-posed operator equations will bepresented: the damped second-order asymptotical regularization methods [1-5] and the fractional-order
asymptotical regularization method [6]. Convergence rate under range type source conditions will be discussed.
Saturation and converse results are given. We will also prove the acceleration phenomena of both methods.
Several numerical examples are presented to show the effectiveness of the proposed new approaches.

References
1. Y. Zhang and B. Hofmann, On the second-order asymptotical regularization of linear ill-posed inverse
problems, Applicable Analysis, DOI: 10.1080/00036811.2018.1517412 (2018).
2. Y. Zhang, R. Gong, X. Cheng and M. Gulliksson, A dynamical regularization algorithm for solving inverse
source problems of elliptic partial differential equations, Inverse Problems, 34, 065001 (2018).
3. R. Gong, B. Hofmann and Y. Zhang, A new class of accelerated regularization methods, with application tobioluminescence tomography, Preprint, arXiv:1903.05972 (2019).
4. G. Baravdish, O. Svensson, M. Gulliksson and Y. Zhang, Damped second order flow applied to image
denoising, IMA Journal of Applied Mathematics, 84, 1082-1111 (2019).
5. Y. Zhang and R. Gong, Second order asymptotical regularization methods for inverse problems in partial
differential equations, Preprint, arXiv:1809.04971 (2018).
6. Y. Zhang and B. Hofmann, On fractional asymptotical regularization of linear ill-posed problems in Hilbert
spaces, Fractional Calculus and Applied Analysis, 22, 699-721 (2019).

116

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MACHINE LEARNING REGULARIZATION OF INVERSE PROBLEM FOR BLACKSCHOLES EQUATION

N. Zyatkova, O. Krivorotkoa and S. Kabanikhinaa
ICM&MG SB RAS, Lavrentiev Avenue, 6, 630090 Novosibirsk, Russia, nikolay.zyatkov@gmail.com

Abstract
The solution of the Black-Scholes equation [1] provides an estimate of the theoretical price of the option
which is a derivative of some underlying asset (for example, stocks). The Black-Scholes model is a multiparameter equation depending on the current price of the underlying asset s that is assumed to follow thegeometric diffusion with Brownian motion Wt

dst   st dt   st dWt ,the strike price K of the option, the time to maturity of the option T expressed in years, the risk free rate r as
well as the price volatility   s  of the underlying asset expressed in years. We consider the Black-Scholesmodel in which the put option price function u  s, T  with the payoff function f  s   max s  K , 0 , whereK  0 , is found as the solution to the direct problem:uu 1 2 2 2u
 sr
 s   s  2  ru, s   0,   , t   0, T  ;
t
s 2
s
u  s, T   f  s  , u  0, t   K .

(1)Volatility is an important value of the underlying asset risk measure used by portfolio managers. However, it
is not possible to clearly observe the volatility of the underlying asset of the option in the financial markets. In
practice, market participants work with the so-called implied volatility   s  from the problem (1) which is
calculated based on the current value of the option observed on the market assuming that this value reflects the
expected risks. The inverse problem for Black-Scholes equation (1) consists in identification of implied volatility
using measured data of option price g  t  at fixed times on the given underlying asset price curve s    t  :u   t  , t   g  t  ,t   0, T  .

(2)

The inverse problem (1) - (2) is ill-posed [2, 3] and reduced to the minimization least square problem that issolved by neural networks with gradient optimization approach [4]. The comparison analysis with otheroptimization methods is presented as well as the short-term prediction of option price behavior.
The work is supported by the Russian Science Foundation (project No. 18-71-10044) and by the RussianFoundation for Basic Research (project No. 18-31-20019).

References
1. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81, 637-654(1973).
2. H. Egger, T. Hein, B. Hofmann, On decoupling of volatility smile and term structure in inverse option pricing,
Inverse Problems, 22, 1247-1259 (2006).
3. V. Isakov, Recovery of time dependent volatility coefficient by linearization, Evolution Equations Control
Theory, 3, 119-134 (2014).
4. D.E. Rumelhart, G.E. Hinton, R.J. Williams. Learning internal representations by error propagation, In:
Parallel Distributed Processing, Cambridge, MA, MIT Press, 1, 318-362 (1986).

117

MINISYMPOSIUM
M11: Inverse Problems for Time-fractional PDEs
Organizer:
Eric Soccorsi, Aix Marseille Université, France, eric.soccorsi@univ-amu.fr
Within years, partial differential equations with fractional derivatives have become very
popular among researchers, owing it not only to their effectiveness in applied scientific modeling but
also to their novel mathematical features.
This mini-symposium is mainly concerned with the mathematical analysis of inverse source
problems and coefficient inverse problems for fractional-order partial differential equations, and itaims for bringing researchers together to present and discuss their latest achievements on these topics.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE SOURCE PROBLEMS FOR TIME-FRACTIONAL DIFFUSION (WAVE)
EQUATIONS

Yikan Liuaa
Research Institute for Electronic Science, Hokkaido University. N12W7, Kita-Ward, Sapporo 060-0812, Japan,
ykliu@es.hokudai.ac.jp

Abstract
Within the last decade, evolution equations with fractional derivatives have gathered considerable
attention among both theoretical and applied disciplines due to their feasibility in modeling physical processes
such as anomalous diffusion. In this talk, we consider initial-boundary value problems for time-fractional
diffusion(-wave) equations represented by

wheredenotes the Caputo derivative andinvestigate the following two inverse source problems.
1.
2.

is a symmetric elliptic operator. In this talk, we

Fix. Provided that is known, determine by the single point observation of at.
Fix a nonempty subdomain. Provided thatis known, determineby the partial interior
observation of in.

The uniqueness results of 1 and 2 were first established in [5] and [1] based on the strong positivity property
and the weak vanishing property of the operator, respectively. There are also researches on thecorresponding numerical reconstructions and we refer to [3] as a comprehensive survey on the above inverse
source problems before 2019.
In recent years, we further investigate Problem 1 and generalize the formulation on the following two
directions:(1) reducing the observation data as much as possible, and(2) relaxing the observation period to an arbitrary time interval
The direction (1) is interesting from both theoretical and practical aspects. Meanwhile, the direction (2)provides great freedom in performing observation, which is essentially important in unexpected accidents. The
second part of this talk is devoted to the brief explanation of the latest progress concerning the uniqueness on theabove directions as well as the key ingredients of their proofs (see [2,4]).

References[1] D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source
problem for time-fractional diffusion-advection equations, Inverse Problems, 33, 055013 (2017).
[2] Y. Kian, Y. Liu and M. Yamamoto, Uniqueness of inverse source problems for general evolution equations,
Commun. Contemp. Math., accepted, arXiv:2105.11987.
[3] Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial differential
equations, Handbook of Fractional Calculus with Applications. Volume 2: Fractional Differential Equations,
De Gruyter, Berlin, 2019, 411–430.
[4] Z. Li, Y. Liu and M. Yamamoto, Inverse source problem for a one-dimensional time-fractional diffusion
equation and unique continuation for weak solutions, submitted, arXiv:2112.01018.
[5] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an
application to an inverse source problem, Fract. Calc. Appl. Anal., 19, 888-906 (2016).

119

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A UNIFIED FRAMEWORK FOR THE REGULARIZATION OF FINAL VALUE TIMEFRACTIONAL DIFFUSION EQUATION

Walter Simo Tao Lee
Université Toulouse 1 Capitole,2 rue du doyen Gabriel Marty, 31000 Toulouse, France
wsimotao@math.univ-toulouse.fr

Abstract
My talk focuses on the regularization of backward time-fractional diffusion problem on unbounded domain.
This problem is well-known to be ill-posed, whence the need of a regularization method in order to recoverstable approximate solution. For the problem under consideration, I present a unified framework of
regularization which covers some techniques such as Fourier regularization [3], mollification [2] andapproximate-inverse [1]. I investigate a regularization technique with two major advantages: the simplicity ofcomputation of the regularized solution and the avoid of truncation of high frequency components (so as to avoid
undesirable oscillation on the resulting approximate-solution). Under classical Sobolev-smoothness conditions, I
derive order-optimal error estimates between the approximate solution and the exact solution in the case where
both the data and the model are only approximately known. In addition, an order-optimal a-posteriori parameterchoice rule based on the Morozov principle is given. Finally, via some numerical experiments in twodimensional space, I illustrate the efficiency of our regularization approach and we numerically confirm thetheoretical convergence rates established in the paper.

References[1]. A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind, Inverse
Problems, 6(3):427, 1990.
[2]. N. Van Duc, P. Q. Muoi, and N. Van Thang, A mollification method for backward time-fractional heat
equation, Acta Mathematica Vietnamica, 45(3):749--766, 2020.
[3]. M. Yang and J. Liu, Fourier regularization for a final value time-fractional diffusion problem, Applicable
Analysis, 94(7):1508--1526, 2015.

120

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEMS FOR FRACTIONAL DIFFUSION EQUATION WITH ONE
MEASUREMENT

Lauri Ylinen
Department of Mathematics and Statistics, University of Helsinki, Finland, Pietari Kalmin katu 5, 00014
University of Helsinki, Lauri.Ylinen@Helsinki.fi

Abstract
We consider an inverse problem for the space-time fractional diffusion equation

where

is a complete Riemannian manifold without boundary,is the Caputo derivative of order,is the Laplace-Beltrami operator on , and β ∈ (0,1]. We show that it is possible to construct asource that is supported on, where is a given open subset of the manifold and, so that the
manifold
is determined up to a Riemannian isometry by the evolution of the corresponding solution on.
This is joint work with Tapio Helin, Matti Lassas, Reed Meyerson, and Zhidong Zhang.

121

The 10th International Conference ”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

NUMERICAL ESTIMATION OF A DIFFUSION COEFFICIENT IN SUBDIFFUSION

Zhi Zhou
The Hong Kong Polytechnic University, Hong Kong
zhizhou@polyu.edu.hk

Abstract
In this work, we consider the numerical recovery of a spatially dependent diffusion coeffcient in a
subdiffusion model from distributed observations. The subdiffusion model involves a Caputo fractional
derivative of order   (0,1) in time. The numerical estimation is based on the regularized output leastsquares formulation, with an H () penalty. We prove the well-posedness of the continuous formulation,e.g., existence and stability.
Next, we develop a fully discrete scheme based on the Galerkin finite element method in space andbackward Euler convolution quadrature in time. We prove the subsequential convergence of the sequence of
discrete solutions to a solution of the continuous problem as the discretization parameters (mesh size and time
step size) tend to zero. Further, under an additional regularity condition on the exact coeffcient, we derive
convergence rates in a weighted norm for the discrete approximations to the exact coeffcient. The analysis relies
heavily on suitable nonstandard nonsmooth data error estimates for the direct problem.
We provide illustrative numerical results to support the theoretical study.
1

122

MINISYMPOSIUM
M12: Electrical Impedance Tomography: Theory and Applications
Organizers:
Giovanni S. Alberti, University of Genoa, Italy, giovanni.alberti@unige.it
Matteo Santacesaria, University of Genoa, Italy, matteo.santacesaria@unige.it
Electrical impedance tomography (EIT) is an imaging modality that consists in thedetermination of the electrical conductivity distribution inside a body from current and voltage
measurements on its boundary. Applications include medical imaging, nondestructive testing andgeophysical prospecting. Its mathematical formulation was proposed by A.P. Calderon in 1980 and it
has triggered a huge amount of research since then. On the theoretical side, the main issue has been to
prove uniqueness of the related inverse problem, namely, the injectivity of the measurement or
forward map. Concerning applications, EIT faces major numerical hurdles, since the problem isseverely ill-posed. In order to mitigate this instability, strategies ranging from regularization methods
to compressed sensing and machine learning have been employed.
In this minisymposium, we are gathering experts of EIT and of the Calderon's problem toshare recent theoretical and numerical/applied insights.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

LIPSCHITZ STABLE DETERMINATION OF POLYGONAL AND POLYHEDRALCONDUCTIVITY INCLUSIONS FROM BOUNDARY MEASUREMENTS

Elena Beretta
NYU Abu Dhabi, UAE;
Politecnico di Milano, Italy, eb147@nyu.edu; elena.beretta@polimi.it

Abstract
We will present some recent results obtained in collaboration with Elisa Francini and Sergio Vessellaconcerning global Lipschitz stability estimates for the Hausdorff distance of polygonal and polyhedral
conductivity inclusions in terms of the Dirichlet-to-Neumann map

124

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MACHINE LEARNING APPROACH FOR STROKE DETECTION IN ELECTRICALIMPEDANCE TOMOGRAPHY
Valentina Candiania and Matteo SantacesariabaDepartment of Mathematics, University of Genova, via Dodecaneso 35, 16146 Genova, Italy, d,candiani@dima.unige.it;b
Department of Mathematics, University of Genova, via Dodecaneso 35, 16146 Genova, Italy,
matteo.santacesaria@unige.it

Abstract
Reconstruction of images in electrical impedance tomography (EIT) requires the solution of a nonlinear
inverse problem on noisy data. This problem is typically ill-conditioned and solution algorithms need either
simplifying assumptions or regularization based on a priori knowledge. This work considers a different strategy,
it aims at training some classification learners (SVM and Neural Networks) with absolute EIT measurements ona head model [1], in order to binary classify the type of stroke occurred: haemorrhagic, if there is a blood leak inthe brain tissue, or ischaemic, when a blood clot stops the blood supply to a certain area of the brain. The
computed measurements take into account different kinds of possible errors in the measurement setup: slightvariations in the background conductivity and in the contact impedances, misplaced electrodes and mismodeledhead shape.
Classification learners have been trained both in the basic case of one single layer of conductivity(meaning only brain conductivity taken into account) and with the insertion of a skull and a skin layer(respectively less than and as conductive as the brain tissue).
Results show reasonably high probability of detecting the correct type of stroke.

References
1. V. Candiani, A. Hannukainen and N. Hyvönen, Computational framework for applying electrical impedance
tomography to head imaging, SIAM J. Sci. Comput., 41(5), B1034-B1060 (2019).

125

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

THE CALDERÓN PROBLEM WITH LIPSCHITZ CONDUCTIVITIESPedro Caro
BCAM, Alameda Mazarredo 14, 48009 Bilbao, Spainpcaro@bcamath.org

Abstract
We will discuss the Calderón problem for Lipschitz conductivities. In particular, we will recall theuniqueness question, and we will discuss the difficulties to produce a reconstruction algorithm.

References
1. Caro, P., & Rogers, K. (2016). Global Uniqueness For The Calderón Problem with Lipschitz Conductivities.
Forum of Mathematics, Pi, 4, E2. doi:10.1017/fmp.2015.9

126

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

GLOBAL CONVERGENCE AND STABLE INVERTIBILITY FOR A ROBIN
TRANSMISSION PROBLEM WITH FINITELY MANY MEASUREMENTS

Bastian Harrach
Department of Mathematics, Goethe University Frankfurt, Germany
harrach@math.uni-frankfurt.de

Abstract
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence ofNewton's method for finite dimensional inverse problems with a continuously differentiable, pointwise convex
and monotonous forward function. Our criterion merely requires evaluating the directional derivative of theforward function at finitely many evaluation points and finitely many directions.
We apply our result to an inverse Robin transmission problem with finitely many measurements that ismotivated by EIT-based corrosion detection techniques. Using a relation to monotonicity and localized potentials
arguments, we will show that the discretized inverse Robin transmission problem always fulfills our criterion if
enough measurements are being used. Thus, our result enables us to determine those boundary measurementsfrom which an unknown coefficient can be uniquely and stably reconstructed with a given desired resolution by
a globally convergent Newton iterationIn discussing weak unique continuation of our operator, a main feature of
our argument relies on
a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthermore, we also
discuss constructive single measurement results based on the approximation and unique continuation propertiesof the equation.

References
1. B. Harrach, H. Meftahi: Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission
Problem, SIAM J. Appl. Math. 79 (2), 525–550, 2019.
2. B. Harrach: On stable invertibility and global Newton convergence for convex monotonic functions,
arXiv:1907.02759, 2019.

127

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

THE INVERSE CONDUCTIVITY PROBLEM FOR COMPLEX CONDUCTIVITIESWITH REGULAR JUMPS

Ivan Pombo
CIDMA, Universidade de Aveiro, Aveiro, Portugal
ivanpombo@ua.pt

Abstract
In this talk we will introduce some new ideas for the inverse conductivity problem in the case where theconductivity is complex and has a regular jump [1]. This material has never been considered in literature and thebest result assumed Lipschitz complex conductivities.
For the study of this problem we model it as an interior transmission problem. We will introduce several
new concepts that were required to treat this problem. One of them is the set of admissible points, which isessential for us since it permits the enlargement of the set of CGO waves. This will be the key point to obtain areconstruction formula for the conductivity in this set of points. Even being apparently a rather weak result, itcannot possibly be obtained by any previous technique, at least that we know of. Therefore, these ideas represent
a new step in this direction and due to being early results we will present some of the footwork necessary toproceed further.
Future ideas are to establish this approach in three dimensions by usage of quaternionic analysis.

References
1. Ivan Pombo, CGO-Faddeev approach for Complex Conductivities with Regular Jumps in two dimensions,
Inverse Problems, 36(2) (2019).

128

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INTERIOR DECAY OF SOLUTIONS TO ELLIPTIC EQUATIONS

Luca Rondi
University of Milan , Italyluca.rondi@unimi.it

Abstract
We investigate the decay in the interior of solutions to elliptic equations. We show that the decay
depends on the properties of the corresponding boundary data, in particular the decay rate is related to the socalled frequency of the boundary datum. This result has interesting applications to the choice of optimalmeasurements for electrical impedance tomography.
This is a joint work with Michele Di Cristo.

129

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

GENERATIVE MODELS FOR ELECTRICAL IMPEDANCE TOMOGRAPHY

Silvia Sciutto
Department of Mathematics, University of Genova, Italy
silvia.sciutto@edu.unige.it

Abstract
Electrical impedance tomography (EIT) is an imaging modality which consists in the determination of
the conductivity distribution of a body from electrode measurements acquired on its boundary. The inverse
problem is severely ill-posed, which results in very low resolution reconstructions and high instability withrespect to noise and modelization errors. In recent years, several machine learning approaches have been able tomitigate the ill-posedness of EIT. Most of these results rely on training datasets with rather specific structure.
In this talk we will present a new data-driven approach based on generative models, which allows us toconsider training datasets with higher variability. Taking inspiration from well-known architectures (e.g. U-Net),
we construct and explicitly characterize a class of injective generative models defined on infinite-dimensional
functions spaces. After an off-line training of the generative model, the proposed reconstruction method consists
in an iterative scheme in the low-dimensional latent space. The main advantages are the faster iterations and the
reduced ill-posedness, which is shown with new Lipschitz stability estimates.

References
1. G. S. Alberti, Á. Arroyo, M. Santacesaria: Inverse problems on low-dimensional manifolds,
arXiv:2009.00574, 2020.
2. G. S. Alberti, M. Santacesaria, S. Sciutto: Continuous Generative Neural Networks. Manuscript in
preparation.

130

MINISYMPOSIUM
M13: Inverse Problems in Geomathematics and Seismology
Organizers:
Andrea Aspri, Universita di Pavia, Italy, andrea.aspri@unipv.it
Elena Beretta, NYU, Abu Dhabi, eb147@nyu.edu
Joonas Ilmavirta, University of Jyväskylä, Finland, joonas.ilmavirta@jyu.fi
Anna Mazzucato, Penn State, USA, alm24@psu.edu (n-59)
Darko Volkov, Worcester Polytechnic Institute, USA, darko@wpi.edu
This minisymposium is concerned with mathematical inverse problems arising fromgeophysics and seismology. An important application that we want to address is the fault
reconstruction problem in seismology: the geometry and the position of the fault are unknown, and theimpulse on the fault producing measurable surface displacements is also unknown. This
minisymposium will bring together analysts, geophysicists with direct experience in seismic modelingand data processing, and investigators who worked on related computational methods.
This minisymposium will spawn discussions on models as well as mathematical andcomputational tools– and together we will identify future research directions.

131

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A STABILITY ESTIMATE FOR AN INVERSE ELECTROSEISMIC PROBLEM

E. Bonnetiera, F. Trikib, Qi XUebaInstitut Fourier, Université Grenoble-Alpes, CS40700, 38058 Grenoble,France, 62 eric.bonnetier@univgrenoble-alpes.fr;b
LJK, Université Grenoble-Alpes, 700 Av. Centrale, 38400 St. Martin d’Hères, France,Faouzi.Triki@univgrenoble-alpes.fr, Qi.Xue@univ-grenoble-alpes.fr

Abstract
In porous media, the motion of charges at the fluid/solid interfaces induces a coupling between seismicand electromagnetic waves. This phenomenon, known as the electro-kinetic effect is at the heart ofelectroseismic imaging, a technique used in oil prospection to image sedimentary layers of porous media. It
combines the high sensitivity to material parameter contrast of electromagnetic waves and the ability to carryinformation with little distortion of seismic waves.
The system we study was proposed by Pride [1]. It couples the Biot equations, which govern the
propagation of elastic waves through porous media, and the Maxwell equations in a bounded domain in R3. As
the electo-kinetic coupling is weak, we only consider the EM to seismic wave conversion in which no coupling
term appears in the equations for the electromagnetic fields (electroseismic model). The objective is to recover
the electromagnetic parameters of the medium and the electrokinetic coupling constant from measurements ofthe electric fields and of the dispacements of the solid and the fluid in a neighborhood of the boundary.
For the electroseismic system, assuming that the physical parameters are sufficiently smooth, we prove a
Carleman estimate, that induces the Hölder stability of the inverse problem [2].

References
1. Steve R Pride. Governing equations for the coupled electromagnetics and acoustics of porous media. PhysicalReview B, 50(21):15678, 1994.
2. E. Bonnetier, F. Triki and Qi Xue. An inverse problem for an electroseismic model describing the coupling
phenomenon of electromagnetic and seismic waves, Inverse Problems 35-045002 (2019).

132

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

EARTHQUAKE INUCLEATION: DIRECT AND INVERSE PROBLEMS
M. Campilloa, V. Canela, I.R. Ionescub and D. Volkovc
a
ISTerre, Université Grenoble Alpes- CS40700 - 38058 Grenoble, France,ichel.campillo@univgrenoble-alpes.fr, canelvince@gmail.comb
LSPM, Sorbonne-Paris-Nord University, 99 Av. JP Clemenet, 93430 Villetaneuse, France,
ioan.r.ionescu@gmail.com
cWorcester Polytechnic Institute, Worcester, MA 01609, USA, darko@wpi.edu.

Abstract
By slow event, we mean important slip taking place on an intermediate time scale (i.e. minutes tomonths). This is much longer than seismic time scales (seconds) but much shorter than geological time scales(hundreds of years). Since slow slip events are aseismic (i.e., there is no associated seismic wave), their detection
is very difficult. Two types of phenomena can be related to slow slip events: silent earthquakes and nucleation(or initiation) phases for (ordinary) earthquakes. The earthquake nucleation (or initiation) phase, which precedes
dynamic rupture, was uncovered by detailed seismological observations and important physical properties of thenucleation phase (characteristic time, critical fault length, etc.) were obtained through simple mathematicalproperties of unstable evolution (see for instance [1]). Early detection of the nucleation phase from surface
displacements has the potential to play a key role in short time prediction of large earthquakes.
Second, we focus on the material damage associated to slow slip events. The existence of thisdamagedzone is traditionally evidenced at depth by seismic tomography that indicates a lower velocity zone
extending several kilometers in depth. A noise based passive monitoring allows to envision a long-term
continuous monitoring of the seismic velocity. Examples of these changes are now well documented and it ispossible to detect them continuously with an unprecedented precision reaching a few 10-5 for relative velocitychanges and very short time resolution. An important issue is to explain how very small macroscale deformation,
associated to the tectonic deformation, could induce observable velocity drop. The study of the wave velocity
drop through a numerical experiment with cohesive discrete media show a large sensitivity due to strain field
heterogeneity [3]. Moreover, wave velocities monitoring in these media can measure subtle processes such as
earthquake initiation phases. Remarkably, our simple model indicates a large “susceptibility” as that observed inreal data.

References
1. M. Campillo and I. R. Ionescu, Initiation of antiplane shear instability under slip dependent friction, J.
Geophys. Res. 122 20363–71 (1997)
2. I. R. Ionescu and D. Volkov, Detecting tangential dislocations on planar faults from traction free surface
observations, Inverse Problems 25, 015012 (2009)
3.V. Canel, M. Campillo, X. Jia and I. R. Ionescu, Damage and wave velocity drop in cohesive granularmaterials: numerical modeling and geophysical implications, submitted (2022)

133

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

COUPLED FLOW AND GEOMECHANICS IN FRACTURED POROUS MEDIUM
Kundan Kumar
University of Bergen, Norway
Kundan.Kumar@uib.no

Abstract
Numerous applications of subsurface engineering involve injection and extraction of fluids. Examples
include geothermal energy extraction, nuclear waste storage, carbon sequestration, petroleum engineering
applications, and energy storage. These anthropogenic activities involve a complex set of processes
involving flow, thermal, chemical reactions, and mechanical effects all possibly coupled to each other. These
complex sets of processes interact with the complex geology that involves ubiquitous fractures and faults. The
network of fractures forms the primary conduit of flow and transport and furthermore, act as the most vulnerable
regions for mechanical instability. The interaction of processes and the complex geometry of fractures brings
computational and mathematical challenges in the simulation of these processes. The fractured medium isgenerally anisotropic, heterogeneous, and has substantially discontinuous material properties spanning several
orders of magnitude.
Our objective is to study coupling of flow and geomechanics in a fractured porous medium setting.
We present a mixed dimensional model for a fractured poro-elasic medium. The fracture is a lower dimensionalsurface embedded in a bulk poro-elastic matrix. The flow equation on the fracture is a Darcy type model that
follows the cubic law for permeability. The bulk poro-elasticity is governed by fully dynamic Biot equations.
The resulting model is a mixed dimensional type where the fracture flow on a surface is coupled to abulk flow and geomechanicsmodel.
There are two directions in which our work contributes to. The first is in extending Biot equations toinclude fracture flow model and complex friction and contact mechanics. The second is in considering different
time schemes for the Multiphysics modelling. We consider finer time steps for the flow and coarser time steps
for the mechanics.
This is joint work with Tameem Almani (Aramco), Maarten de Hoop (Rice), Vivette Girault (Paris
6), and Mary F Wheeler (Austin), Ruichao Ye (Chevron).

134

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DEEP LEARNING ARCHITECTURES FOR NONLINEAR OPERATOR FUNCTIONSAND INVERSE PROBLEMS FOR WAVE EQUATIONS
Matti Lassasa, Maarten de Hoop b, Christopher Wongba
Department of Mathematics and Statistics, University of Helsinki, Finland;b
Department of Computational and Applied Mathematics, Rice University, USA

Abstract
We consider a new type of deep neural network developed to solve nonlinear inverse problems. In
particular, we consider inverse problems for a wave equation where one want to determine an unknown wave
speed from the boundary measurements. In particular, we consider the model where the wave propagation isgoverned by the linear acoustic wave equation on an interval. A novel feature of the studied neural network isthat the data itself form layers in the network. This corresponds to the fact that data for the inverse problem is alinear operator that maps the boundary source to the boundary value of the wave that is reflected from theunknown medium. Even though the wave equation modelling the waves is linear, the inverse problem of finding
the coefficients of this equation is non-linear. Using the classical theory of inverse problems we design a neuralnetwork architecture to solve the inverse problem of finding the unknown wave speed. This makes it possible torigorously analyze the properties of the neural network.
For inverse problems, the main theoretical questions concern uniqueness, range characterisation,
stability and the regularisation strategies for the inverse problems. We will discuss the question when a solution
algorithm generalises from the training data, that is, when the solution algorithm trained with a finite number ofsamples can solve the problem with new inputs that are not contained in the training data. This can viewed as a
new question for classical inverse problems that takes its motivation from machine learning.

References
1. M. de Hoop, M. Lassas, C. Wong: Deep learning architectures for nonlinear operator functions and nonlinear
inverse problems. To appear in Mathematical Statistics and Learning. Preprint: arXiv:1912.11090 (2019)135

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARIZATION FOR SEISMIC SOURCES INVERSION FROMINTERFEROMETRIC DATA

Laurent Seppecher
Institut Camille Jordan, École Centrale de Lyon, France
laurent.seppecher@ec-lyon.fr

Abstract
In this talk, we deal with the inverse source problem from interferometric measurements. Interferometricdata are made of correlations of the wavefield recorded at neighboring positions and frequencies. In someconfigurations, these correlations data are very stable with respect to phase shift due to some wave-speed
uncertainties inside the medium. Hence, trying to recover the source from this data is of large interest. Some wellknown works describe techniques of “interferometric migration”, inspired by the classic back-propagation
technique, to form an image of the source from this data. A refocalization phenomenon is observed and the built
image can be much better that the classic back-propagation of the wavefield measurements.
The inverse problem from interferometric measurements is challenging as the system of equations to solve
becomes a quadratic system. In the sense of least squares, the objective function is then a non convex 4-th orderpolynomial.
We will show that it is possible to solve this problem using an additional regularization term and how someerror estimates can be proved. In various numerical applications, we will illustrate the stability of thereconstruction with respect to some important medium uncertainties leading to an important phase shift. This
technique as a wide potential in other fields such as signal and image denoising problems or the phase retrieval
problem.

136

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STABILITY PROPERTIES FOR A CLASS OF INVERSE PROBLEMS WITH
APPLICATIONS TO NEURAL NETWORK SOLUTIONS

Darko Volkov
Worcester Polytechnic Institute, 100 Institute Road,MA 01609 Worcester,USAdarko@wpi.edu

Abstract
We establish Lipschitz stability properties for a class of inverse problems. In that class, the associateddirect problem is formulated by an integral operator
depending non-linearly on a parameterandoperating on a function . In the inversion step bothandare unknown but we are only interested inrecovering . We discuss examples of such inverse problems for the elasticity equation with applications toseismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity andregularity properties for, we prove that the inverse problem with a finite number of data points is solvableand that the solution is Lipschitz stable in the data.
We show a reconstruction example illustrating the use of neural networks

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

BAYESIAN INFERENCE OF SEISMIC EVENTS FROM LOCAL DEFORMATIONSUSING THE WEAKLY-ENFORCED SLIP METHOD
G.J. van Zwietena, E.H. van Brummelenb and R.H. HanssencaEvalf Computing, Burgwal 45, 2611 GG Delft, The Netherlands, gertjanvanzwieten@evalf.com;b
Faculty of Mechanical Engineering, Eindhoven University of Technology, The Netherlands;c
Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands

Abstract
Earthquakes cause lasting changes in static equilibrium, resulting in global deformation fields that canbe observed. Consequently, deformation measurements such as those provided by satellite based InSAR
monitoring can be used to infer an earthquake’s faulting mechanism. This inverse problem requires a numericalmodel that is both accurate and fast, as typical inverse procedures require many forward evaluations. The
Weakly-enforced Slip Method (WSM) [1] was developed to meet these needs, but it was not before applied in an
inverse problem setting. In this contribution we show that the WSM is able to accurately recover slip
distributions in a Bayesian-inference setting.
The WSM is a Galerkin method: given stiffness tensor C , slip vector b and fault plane normal vector
 , the WSM displacement field u , is that which for any test function v satisfies: u : C : v   b  C : v


(1)

F

Crucially, the left hand side of this equation translates to the standard stiffness matrix for linear elastic
computations. This means that expensive steps such as meshing, assembly, and factorization are independent of
the fault geometry and can all be reused. The novelty of the WSM lies in the right hand side, which involves anintegral over the fault plane alone, and as such is relatively cheap to evaluate. The result is a method that has thesame up front costs as an ordinary elasticity computation but comparatively low costs per fault. Compared toconventional solution strategies, the main downside of the WSM is that the solution space cannot account fordiscontinuities of the displacement field. Instead, WSM solutions feature sharp gradients in the vicinity of thefault as approximations of a jump. While this implies a large error local to the fault, this error is shown [1] todecay rapidly with distance, and to converge optimally under mesh efinement.
To study the WSM in an inverse setting we use the following approach. First, we synthesize
deformation data based on analytical solutions for a (2D and 3D) homogeneous half-space and add measurement
noise. Then, we use Bayesian inference to form a posterior probability of the faulting mechanism, using the same
forward model to establish the theoretical optimum. Finally, we repeat the inversion using the WSM forward
model to identify the influence of the WSM’s error components and develop potential mitigating practices. In a
representative scenario, we found a finite element size of 1 km to be sufficiently fine to generate a posterior
probability distribution that is similar to the theoretical optimum. Buried faults of moderate depth required no
special intervention, as the surface is everywhere removed from the dislocation. Rupturing faults, on the other
hand, require a masking zone of 10 km to avoid numerical disturbances that would otherwise be nduced by the
local discretization error. Errors induced by boundary treatment were automatically mitigated by treating thedisplacement data as inherently relative.
Our results demonstrate that the WSM is a viable forward method for earthquake inversion problems.
While our synthesized scenario is basic for reasons of validation, our results are expected to generalize to thewider gamut of scenarios that finite element methods are able to capture. This has the potential to bring modeling
flexibility to a field that is often forced to impose model restrictions in a concession to computability.

References
1. G.J. van Zwieten, E.H. van Brummelen, K.G. van der Zee, M.A. Guti´errez, R.F. Hanssen. Discontinuities
without discontinuity: The Weakly-enforced Slip Method. Computer Methods, Applied Mechanics andEngineering, 2014; 271: 144-166

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MINISYMPOSIUM
M14: Mathematical Methods in Tomography Across the Scales
Organizers:
Peter Elbau, University of Vienna, Austria, peter.elbau@univie.ac.at
Leonidas Mindrinos, University of Vienna, Austria, leonidas.mindrinos@univie.ac.at
Tomographic imaging continues to attract a lot of interest from the inverse problems
community because of the increasing need of new mathematical models describing better theexperiments and of efficient and sophisticated computational algorithms handling big data. In Austria,
five Universities and one institute are collaborating in this direction under the Special Research Project“Tomography across the scales” founded by the Austrian Science Fund (FWF).
In this mini-symposium we bring together researchers, members of the project and external
collaborators, working on imaging problems from the nanoscale of single molecule imaging tomicroscale of multi-modal imaging. We cover topics such as adaptive optics, quantitative
reconstructions, image processing, and integral transforms.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DATA-DRIVEN REGULARIZATION

Andrea Aspri
Department of Mathematics, University of Milan, Via Cesare Saldini 50, 20133Milan, Italy,
andrea.aspri@unimi.it

Abstract
In this talk I will speak about some recent results on the study of linear inverse problems under thepremise that the forward operator is not at hand but given indirectly through some input-output training pairs.
We show that regularisation by projection and variational regularisation can be formulated by using the training
data only and without making use of the forward operator. We will provide some information regarding
convergence and stability of the regularized solutions. Moreover, we show, analytically and numerically, that
regularisation by projection is indeed capable of learning linear operators, such as the Radon transform.
This is a joint work with Yury Korolev (University of Cambridge) and Otmar Scherzer (University of
Vienna and RICAM).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

QUANTITATIVE SEISMIC IMAGING USING RECIPROCITY-BASED METHODSENABLING ARBITRARY PROBING SOURCES

Florian Faucher
University of Vienna, Austria
florian.faucher@univie.ac.at

Abstract
We study the inverse problem associated with the propagation of time-harmonic waves for seismic
applications, where we work with partial reection data. The recovery of the subsurface Earth medium parameters
is conducted using an iterative minimization procedure of a mis_t functional.
Following the measurements of two types of data (e.g., with dual-sensors devices), we can define a new misfit
functional based upon the reciprocity-gap. The main feature of this misfit functional is to allow a separation
between the observational and numerical sources. Namely, the position of the observational sources do not need
to be known, and arbitrary probing sources can be used for the numerical simulations. In particular, it o_ers thepossibility to create adapted computational acquisitions in order to reduce the numerical burden.
We shall present the method for both acoustic and elastic wave problems, and illustrate the robustnessof our approach with respect to shot stacking with three-dimensional experiments.

References[1] F. Faucher, G. Alessandrini, H. Barucq, M. V. de Hoop, R. Gaburro and E. Sincich, Full Reciprocity-Gap
Waveform Inversion in the frequency domain, enabling sparse-source acquisition, Arxiv preprint
arXiv:1907.09163 (2019)[2] G. Alessandrini, M. V. de Hoop, F. Faucher, R. Gaburro and E. Sincich, Inverse problem for the Helmholtz
equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization,
ESAIM: M2AN (2019).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DATA-DRIVEN METHODS IN INVERSE PROBLEMS

Leon Frischauf
Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien,leon.frischauf@univie.ac.at

Abstract
We will focus on data-driven regularization by projection showing some numerical results andcomparisons with different methods. We will also discuss in details “Seidman's non-convergence example on
regularization”. Additionally, we provide an application of a projection algorithm, utilized and applied in frames
theory, as a data driven reconstruction procedure in inverse problems.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A FRAME DECOMPOSITION OF THE ATMOSPHERIC TOMOGRAPHYOPERATOR

Simon Hubmer and Ronny Ramlau
Johann Radon Institute for Computational and Applied Mathematics, Austria, simon.hubmer@ricam.oeaw.ac.at

Abstract
We consider the problem of atmospheric tomography, as it appears for example in adaptive optics
systems for extremely large telescopes. A singular-value-type decomposition of the underlying atmospherictomography operator has been derived previously, for setups with only natural guide stars and square apertureshapes.
We extend these results by deriving a frame decomposition of the atmospheric tomography operator
which allows both mixed natural and laser guide star setups, as well as arbitrary aperture shapes.
The significance of the derived results is discussed both from an analytical and a numerical perspective.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A PROBABILISTIC ORACLE INEQUALITY AND QUANTIFICATION OF
UNCERTAINTY OF A MODIFIED DISCREPANCY PRINCIPLE FOR STATISTICAL
INVERSE PROBLEMSTim Jahn
HCM, Endenicher Allee 62, 53115 Bonn, Germany
jahn@ins.uni-bonn.de

Abstract
In this talk we consider spectral cut-off estimators to solve a statistical linear inverse problem underarbitrary white noise. The truncation level is determined with a recently introduced adaptive method based on theclassical discrepancy principle. We provide probabilistic oracle inequalities together with quantification of
uncertainty for general linear problems. Moreover, we compare the new method to existing ones, namely early
stopping sequential discrepancy principle and the balancing principle, both theoretically and numerically.

References
1. T. Jahn, Optimal convergence of the discrepancy principle for polynomially and exponentially ill-posed
operators under white noise, Numerical Functional Analysis and Optimization, 2021,
https://arxiv.org/abs/2104.06184
2. T. Jahn, A probabilistic oracle inequality and quantification of uncertainty of a modified discrepancy principle
for statistical inverse problems, arxiv preprint, 2022 https://arxiv.org/abs/2202.12596

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM IN AN
ABSORBING AND SCATTERING MEDIUM WITH APPLICATIONS TO OPTICAL
MOLECULAR IMAGING
Kamran Sadiq
Johann Radon Institute Linz, Altenbergerstraße 69, 4040 Linz, Austria
kamran.sadiq@ricam.oeaw.ac.at

Abstract
We revisit the inverse source problem in a two dimensional absorbing and scattering medium andpresent a direct reconstruction method, which does not require iterative solvability of the forward problem, using
measurements of the radiating flux at the boundary. The approach is based on the Cauchy problem for a
Beltramilike equation for the sequence valued maps, and extends the original ideas of A. Bukhgeim from thenon-scattering to the scattering media. Of novelty here, the medium has an anisotropic scattering property that isneither negligible nor large enough for the diffusion approximation to hold.
The numerical realization of the proposed reconstruction method is also presented, which is amenable
for such scattering media. The feasibility of the proposed algorithm is demonstrated in several numericalexperiments, including simulated scenarios for parameters meaningful in Optical Molecular Imaging.
This is joint work with Alexandru Tamasan and Hiroshi Fujiwara.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DISPLACEMENT FIELD ESTIMATION UTILIZING SPECKLE INFORMATION FOR
PARAMETER RECOVERY IN QUANTITATIVE ELASTOGRAPHY

E. Sherinaa, S. Hubmerb, L. Krainzc, O. Scherzerd,e and W. DrexlerfaFaculty of Mathematics, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria,
ekaterina.sherina@univie.ac.at;bJohann Radon Institute Linz, Altenbergerstraße 69, 4040 Linz, Austria, simon.hubmer@ricam.oeaw.ac.at;
cCenter for Medical Physics and Biomedical Engineering, Medical University of Vienna, Wӓhringer Gürtel 1820, 1090 Vienna, Austria, lisa.krainz@meduniwien.ac.at;dFaculty of Mathematics, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria,
otmar.scherzer@univie.ac.at;eJohann Radon Institute Linz, Altenbergerstraße 69, 4040 Linz, Austria, otmar.scherzer@ricam.oeaw.ac.at;fCenter for Medical Physics and Biomedical Engineering, Medical University of Vienna, Wӓhringer Gürtel 1820, 1090 Vienna, Austria, wolfgang.drexler@meduniwien.ac.at;

Abstract
Diseases like cancer and arteriosclerosis often cause changes of tissue stiffness in the micrometer scale.
Thus, it is possible to identify malignant formations inside tissues by means of quantitative elastography. As a
medical imaging modality, elastography examines the tissue by deforming it mechanically to produce qualitative
or desirably quantitative maps of the tissue. In this work, we consider two successive, closely related problems
which arise as we attempt to quantitatively recover the unknown biomechanical parameters of a tissue. First, theinversion method we use for finding the parameters [1] relies on the knowledge of the internal displacement field
which occurs in the tissue due to deformation. In the considered physical experiments, successive OCT/PAT
scans of the object interior are available before and after compression which are typically used as the input forthe displacement estimation. In our case, additional speckle information extracted from the OCT scans is used
for enhancing the quality of the displacement reconstruction.
We present and analyse a framework for including the speckle information in the estimation procedure.
Second, we are interested in restoring elastic material parameters from the obtained displacement field todemonstrate the practical usefulness of the proposed approach with speckle data.
In particular, we present numerical results for the displacement and parameter estimation based on both
simulated and experimental data.

References
1. S. Hubmer, E. Sherina, O. Scherzer and A. Neubauer, Lamé Parameter Estimation from Static Displacement
Field Measurements in the Framework of Nonlinear, Inverse Problems, SIAM Journal on Imaging Sciences,11(2), 1268-1293 (2018).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

QUANTITATIVE OCT RECONSTRUCTIONS FOR DISPERSIVE MEDIA

Leopold Veselka
Faculty of Mathematics, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria,
leopold.veselka@univie.ac.at

Abstract
Optical coherence tomography(OCT) is a non-invasive imaging technique, which produces highresolution images of the inner structure of biological tissues by measuring the intensity of the backscattered lightfrom the sample. In this talk we discuss the inverse scattering problem of extracting quantitative information
about the underlying object in form of reconstructing its optical properties from OCT measurements. We
consider a linear dispersive medium with multi-layered structure, described by a frequency- and depth-dependent
parameter, the electric susceptibility. The parameter identi_cation problem can then be formulated as a onedimensional inverse problem. Furthermore, an iterative scheme, based on the solution of Helmholtz equation,
which is the core of the reconstruction algorithm for the frequency dependent parameter will be also presented.
This is joint work with Peter Elbau and Leonidas Mindrinos.

147

MINISYMPOSIUM
M15: Inverse Source Problems with Applications to Planetary Sciences andMedical Imaging
Organizer:
Laurent Baratchart, INRIA, France, laurent.baratchart@inria.fr
Inverse Poisson problems with source term in divergence form is a class of inverse problems
occuring naturally in Electromagnetics, under quasi-static assumptions. They arise for instance inconnection with Inverse Magnetization Problems, e.g. in Paleomagnetism or Geomagnetism, as wellas in Electro-Encephalography or Magneto-Encephalography. In recent years, tools from harmonic
analysis, in particular extremal problems in Hardy spaces of harmonic gradients, as well as notions ofsparsity from geometric measure theory, have been applied to recovery problems, moment estimation,and source separation.

148

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEM OF SOURCE IDENTIFICATION IN
ELECTROENCEPHALOGRAPHY (EEG)Paul Asensio
Inria Sophia Antipolis, 2004 Route des Lucioles, 06902 Valbonne, France
paul.asensio@inria.fr

Abstract
We aim to solve the problem of source identification in EEG by modeling the sources as R 3 -valued
finite linear combinations of dipoles. To achieve this we formulate the problem over the Banach space of R 3 valued Radon measures, compactly supported in a ball. This problem is ill-posed hence we minimize a Tikhonov
type problem with a "weighted" total variation regularization term, to achieve dipole recovery.
We discuss constructive issues, using an algorithm developed by Bredies and Pikkarainen [1], andsuggest a criterion for an appropriate choice of the "weight" towards a good dipoles recovery.

References[1] K. Bredies and H. Pikkarainen. Inverse problems in spaces of measures. European Series in Applied andIndustrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, 19, 12 2010.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOME INVERSE MAGNETIZATION PROBLEMS IN GEOSCIENCE

Christian Gerhards
TU Bergakademie Freiberg, Germany
christian.gerhards@geophysik.tu-freiberg.de

Abstract
Potential Field Problems, in particular those concerning the Earth’s gravity and magnetic field, have along history in geophysics. Nowadays, corresponding data is available at a variety of scales, ranging frommicroscopy data over surface and airborne data to global satellite data. In this presentation, we want to provide
an overview on some inverse problems arising with magnetic field data, of which further details are discussed inaccompanying presentations.
Topics that will discussed are uniqueness issues for the recovery of magnetic sources in a volumetric aswell as surface setup, separation of magnetic sources, and possibly connections to other geophysical quantities(like geothermal heat flow).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOME MEASURE-THEORETIC ASPECTS OF PLANAR MAGNETIZATIONRECONSTRUCTION

C. Villalobos Guillen
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA,
cristobal.villalobos.guillen@protonmail.com

Abstract
When modeling magnetizations by R3-valued Borel measures, the kernel of the forward operator, mapping
magnetizations supported on the plane {z = 0} to the restriction of the magnetic fields they induce to a
sufficiently dense subset of an analytic surface, consist of divergence free measures whose 3rd component is zero(see [1] and [2]). Using this fact as a motivation, we will show how any divergence free R 2-valued Borel
measure on R2 can be decomposed into line integrals over Jordan curves.
This result has two important implications for the recovery of planar magnetizations. First, that any
magnetization supported on sufficiently separated straight lines has minimal total variation, in the measuretheoretic sense, among all other planar magnetizations that induce the same field. Second, that the group LASSO
regularization problem in this context, which amounts to total variation regularization, has a unique solution forany value of the regularization parameter.

Reference
1. L. Baratchart, D. P. Hardin, and C. Villalobos Guillén. Inverse potential problems in divergence form formeasures in the plane. ESAIM Control Optimisation and Calculus of Variations 27:87 (2021).
DOI:10.1051/cocv/2021082
2. L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington and E. B. Saff. Inverse potential
problems for divergence of measures with total variation regularization. Foundations of Computational
Mathematics, Nov 2019.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RELATING HARDY COMPONENTS OF COMPACTLY SUPPORTEDMAGNETIZATIONS ON SPHERE
Xinpeng Huang
Institute of Geophysics and Geoinformatics, TU Bergakademie Freiberg, Gustav-Zeuner-Str. 12, 09599
Freiberg, Germany,
xinpeng.huang@geophysik.tu-freiberg.de

Abstract
The null space of the magnetic potential with sources supported on a spherical membrane can becharacterized by the Hardy-Hodge decomposition. Moreover, if the underlying magnetization is spatiallylocalized in a subdomain of the sphere, there exists a unique map between its inner and outer Hardy components.
Using layer potentials, we will dicuss the contruction of this map and its relation to Cauchy problems,characterize its domain, and prove some further properties of interest for related bounded extremal problems.
This is joint work with Christian Gerhards and Alexander Kegeles.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A “SIMPLE” APPROACH TO THE MAGNETO-STATIC INVERSE PROBLEM

Alexander Kegeles
TU Bergakademie Freiberg, Germany, alexander.kegeles@geophysik.tu-freiberg.de

Abstract
In geomagnetism, we often encounter the following inverse problem: A magnetic source creates asurrounding magnetic potential; we measure that potential and seek the source magnetization.
In general, this problem is ill-posed. In particular, it has no unique solution on the space of squareintegrable vector fields.
In this talk, we will characterize the non-uniqueness of this problem. We will show that a part of themagnetization can be uniquely recovered, but this part has only limited (geo)physical use. Nevertheless, despite
the ill-posed nature of the problem, we also show that the inversion is unique for a dense subspace of squareintegrable vector fields. This subspace rests upon mild physical assumptions and has, quite literally, a simple
mathematical description.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A LAYER POTENTIAL APPROACH TO FUNCTIONAL AND CLINICAL BRAINIMAGING
P. Asensioa, J.-M. Badierb, J. Leblonda, J.-P. Marmoratc, M. Nemairea,da
FACTAS, Inria Sophia Antipolis-Méditerranée, Valbonne, France,bInstitut de Neurosciences des Systèmes, Aix-Marseille Université, Marseille, France,cCenter of Applied Mathematics, Ecole des Mines ParisTech, Sophia Antipolis,France.
dInstitut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France, masimba.nemaire@inria.fr

Abstract
We consider the inverse source recovery problem from stereo-ElectroEncephaloGraphy (sEEG),
ElectroEncephaloGraphy (EEG) and MagnetoEncephaloGraphy (MEG) point-wise data. We regard this as aninverse source recovery problem for [X]3 -valued vector-fields, where X is a Banach space of source terms
supported on the grey/white matter interface within the brain, which together with the skull and the scalp form a
non-homogeneous layered conductor.
We assume that the quasi-static approximation of Maxwell’s equation holds for the electro-magnetic
fields considered. The electric data is measured point-wise inside (sEEG data) and outside the conductor (EEG
data) while the magnetic data (MEG data) is measured only point-wise outside the conductor. This ill-posed
problem is solved via Tikhonov regularization on triangulations of the interfaces and a piecewise linear modelfor the current on the triangles. Both in the continuous and discrete formulation the electric potential is expressed
as a linear combination of the Newton potential of the (distributional) divergence of the source and double layer
potentials while the magnetic flux density in the continuous case is a vector-surface integral whose discrete
formulation features single layer potentials. A main feature of our approach is that these contributions of thesingle and double layer potentials can be computed exactly, [1]. Regularity conditions for the electric potential in
the inverse source recovery problem allow the associated Cauchy transmission problem to be inadvertently
solved as well. For the latter, we only propagate the electric potential while the normal derivatives (current flux)
at the interfaces of discontinuity of the electric conductivities are computed directly from the resulting solution.
This reduces the computational complexity of the problem. Because of the connection between the magnetic flux
density and the electrical potential in conductors such as the one we explore, a coupling of the sEEG, EEG and
MEG data for solving the respective inverse source recovery problems simultaneously is direct, [2]. We treat
these problems in a unified approach that uses single and/or double layer potentials. To solve the associatedTikhonov problems we use an alternating minimisation procedure that alternates between refining the source
localisation and the cortical mapping, [3].
Numerical experiments were performed using meshes of realistic head geometries with synthetic data
and the results will be discussed..

References
1. S. Nintcheu Fata. Explicit expressions for 3D boundary integrals in potential theory. International Journal for
Numerical Methods in Engineering, 78:32–47, 2009.
2. J. Sarvas. Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Physics inmedicine and biology, 32 1:11–22, 1987.
3. J. Both. On the rate of convergence of alternating minimization for non-smooth non-strongly convex
optimization in Banach spaces. Optimization Letters, pages 1–15, 05 2021.

154

MINISYMPOSIUM
M16: Recent Advances in Inverse Problems and Distributed Parameter Systems
Minisymposium in honor of Professor H. T. Banks' long career in appliedmathematics and his leadership in inverse problems
Organizers:
John A. Burns, Interdisciplinary Center for Applied Mathematics, Virginia Tech., USA,jaburns@vt.edu
Fumio Kojima, Kobe University, Japan, kojima@koala.kobe-u.ac.jp
Since the middle of the 1970’s, Professor H. Thomas Banks, has been one of the world’s mostdistinguished and active researchers in the area of parameter identification and inverse problems forpartial differential equations. Mathematical topics to which H.T. Banks has made contributions (manymotivated by applied problems) include: control of delay and partial differential equations,approximation and computational methods, identification and inverse problem methodology for linear
and nonlinear distributed parameter systems, nonlinear damping and hysteresis modeling,homogenization, and the mathematics of smart material structures. Service to the profession has
always been an important obligation for H.T. Banks. He has received generous recognition for his
service and research efforts with students, postdocs and colleagues. These honors include Professor
Honoraire, Universite de Compaigne, 1977; appointed University Professor and Drexel Professor of
Mathematics, NCSU, 1992; elected Fellow, IEEE; Fellow, Institute of Physics,; Fellow, SIAM;
Fellow, AAAS; IEEE-CSS Control Systems Technology Award, 1996; Distinguished Alumni Award,
Purdue Univ., 1998; named Alumni Distinguished Graduate Professor, NCSU, 2000. He has served on
more that 20 editorial boards over the past 20 years.
In this minisymposium we will recognize his accomplishments with presentations by a varietyof scientists on topics which are connected to and influenced by the pioneering work of Professor
Banks.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FINITE DIFFERENCE SCHEMES FOR A STRUCTURED COAGULATIONFRAGMENTAITON MODEL IN THE SPECE OF MEASURES

Azmy S. Ackleh a, Rainey Lyons a, Nicolas Saintier baDepartment of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, U.S.A.,
azmy.ackleh@louisiana.edu and rainey.lyons1@louisiana.edu;b
Departamento de Matematica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428)
Pabellon I - Ciudad Universitaria - Buenos Aires - ARGENTINA; nsaintie@dm.uba.ar

Abstract
We formulate a structured coagulation-fragmentation model on the space of Radon measures equippedwith the bounded Lipschitz norm. This formulation unifies the study of discrete and continuous coagulationfragmentation models under one framework. We prove that the model is well-posed and show that it can reduce
down to the classical discrete and continuous coagulation-fragmentation models. To understand the interplay
between the physical processes of coagulation and fragmentation and the biological processes of growth,
reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary
solutions are absolutely continuous under some conditions on model parameters. We then develop and compare
multiple finite difference schemes. In particular, we develop a fully explicit scheme, a semi-implicit scheme, andan explicit scheme based on the mass conservation law governed by the model. We prove convergence for eachscheme and test the performance of the schemes against multiple examples.

References[1] A.S. Ackleh, R. Lyons, and N. Saintier. Structured coagulation-fragmentation equation in the space of radonmeasures: Unifying discrete and continuous models. ESAIM: Mathematical Modelling and Numerical
Analysis, in press.
[2] A.S. Ackleh, R. Lyons, and N. Saintier. Finite difference schemes for a structured
population model in the space of measures. Mathematical Biosciences and Engineering: MBE, 17:747 775,
2020.
[3] A.S. Ackleh, R. Lyons, and N. Saintier. Finite difference schemes for structured coagulation-fragmentation
equation in the space of radon measures. submitted.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

POPULATION DYNAMICS IN APPLIED ECOLOGY: MODELS & EXPERIMENTSJohn E. Banks
California State University, Monterey Bay, USA
jebanks@csumb.edu

Abstract
Understanding how and why populations fluctuate is a central concern in applied ecology.
Mathematical models, especially those parameterized with field- and laboratory-derived data, are a powerful toolfor understanding population dynamics and the forces governing population regulation. I describe several
examples taken from nearly two decades of collaborative research with the H.T. Banks lab in which field/lab
data from ecological experiments were incorporated into population models in order to estimate parameters andderive insights into drivers of population conservation or control.
Case studies taken from ecological field studies include research around issues of forest birdconservation as well as ecosystem services such as pest control and pollination in agricultural systems.
Finally, I address the overall utility of using inverse-problem approaches for parameter estimation alongwith delay differential equations to better understand population dynamics in applied ecology.

157

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

NUMERICAL METHODS FOR CONTROL, OPTIMIZATION, DESIGN AND
ESTIMATION OF INFINITE DIMENSIONAL SYSTEMSJohn A. Burns
Interdisciplinary Center for Applied Mathematics, Virginia Tech,Blacksburg, VA 24061-0531
jaburns@math.vt.edu

Abstract
This study is motivated by sensor location problems for infinite dimensional linear distributed
parameter systems (DPS) defined by partial and delay differential equations (PDEs). The term “sensor location"
usually refers to a spatial location. The problem of “optimal" sensor (and actuator) location plays an importantrole in controller design, system identification and state estimation and, depending on the application, can beformulated in many ways. Any optimal location problem implies that there
is a cost function to be minimized and the choice of this cost function places specific requirements on thecomputational methods needed to solve the problem. In addition, choices such as “optimizing observability" canlead to ill-posed problems and result in non-convergence of suboptimal placements. On the other hand, minimum
error variance methods (Kalman filtering) can be shown be well-posed. Efficient computational methods forsolving the corresponding Riccati partial differential equation have been developed and are readily available.
In this presentation we focus on state estimators based on the Luenberger observer and suggest somenatural cost functions for sensor location. The framework is fairly general and applies to both finite and infinite
dimensional systems.
Examples are given to illustrate the ideas.

158

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

WHY ARE THERE SO MANY STRUCTURAL HEALTH MONITORING
ALGORITHMS?
Daniel J. Inman
University of Michigan, USA
daninman@umich.edu

Abstract
H.T Banks taught me the importance of existence and uniqueness in trying to solve damage detection
problems which later become the area of structural health monitoring. It often happens in science, engineeringand mathematics that different discipline specific researchers ignore problems and techniques in those disciplines
not directly related to their own.
The purpose of this review recognize the important similarities between various disciplines and thatthese similarities can be exploited to provide new results. Specifically some forms of structural healthmonitoring are very much like several other disciplines. In order to limit the scope of this presentation it islimited to structures describing linear structural systems that can be well defined in second order form resulting
from the direct application of Newton’s law. Four problems are considered that are mathematically identical.
They are: inverse eigenvalue problems from mathematics, eigenstructure assignment from controls, model
updating (also called model correction) from structural dynamics and structural health monitoring, largely fromthe fields of structural dynamics and signal processing.

159

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEM FOR ELECTROMAGNETIC PROPAGATION IN HUMAN
MUSCLE TISSUES USING HAMILTON MONTE CARLO METHOD

Fumio Kojima
Professor Emeritus, Kobe University, Japankojima@koala.kobe-u.ac.jp

Abstract
Recently, electromagnetic imaging is of considerable interest in the signal detection and theidentification related to the development of new ceramic materials, the early detection of anomalies in polymer,
metabolic functioning of human bodies, etc [1]. We consider an inverse problem arising in human muscle tissues
based on frequency dependent dielectric media using Debye formulation. Let c  (  0 0 ) 0.5 be the speed oflight and let ki  i /(2c) be the interrogation wavenumber with respect to the specified angularfrequency i . Setting as 0  0 /  0 , the complex permittivity is represented byˆr (ki )    
whereMi 0 0
 m
,

2ki
m1 1  i ( 2c m ) ki

 m  ( s    ) g m .

(1)

(2)

The reflective coefficient r at the plane interface between air and a dielectric medium is given byr (ki ) 11

ˆr ( ki ),ˆr ( ki ) i(3)Thus, measurements by time domain reflectometry method (TDR) provide the observable reflectance given by

R(ki )  r (ki ) r (ki ) , for i  0,1, ..., N  1(4)

Our inverse problem treated here is to identify the unknown parameter vectorq M  k0 ,   ,  s ,   ,  k ,  k k 1  R M qM =Kin (1)-(3) based on the model output (4).
An estimation algorithm can be performed by sampling procedures for the posteriori distribution
from which sample paths can be drawn using Markov chains. In this paper, we demonstrate an estimation
scheme using Hamiltonian Monte Carlo method (HMC) that is a Metropolis method making use of gradient
information to reduce random walk behavior [2].

References
1. H. T. Banks, M.W. Buksas and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces
and Acoustic Wavefronts, Frontier in Applied Mathematics, Vol. FR21, SIAM, Philadelphia 2000.
2. R.M. Neal, MCMC using Hamiltonian Dynamics, Handbook of Markov Chain Monte Carlo, Eds. S. Brooks,
A. Gelman, G. Jones, and X. Meng, Chapman & Hall, CRC Press, London, 2011.

160

MINISYMPOSIUM
M17: Parameter Identification Problems for PDEs: Theoretical andComputational Aspect
Organizer:
Tran Nhan Tam Quyen, Georg-August-University of Goettingen, Germany, quyen.tran@unigoettingen.de
Many phenomena in real-life are modeled by partial differential equations. Roughly theyillustrate the relationships between result states, processes concerning derivatives and relatedparameters. Nevertheless, it might happen that some models are imprecise in the practical setting:parameters such as coefficients, source terms, boundary conditions may be subject to uncertainty
which must be identified from measurement data of the states.
The minisymposium is a forum for scientists to present recent results on the uniqueness,stability and convergence of regularization methods for the parameter identification problem as well as
on reconstruction algorithms and experimental implementations.

161

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECONSTRUCTION OF LAMÉ PARAMETERS IN LINEAR ELASTICITYSarah Eberle
Goethe-University of Frankfurt am Main, Germany
eberle@math.uni-frankfurt.de

Abstract
The main motivation of this problem is the non-destructive testing of elastic structures for material
inclusions. We consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-toDirichlet map.
We show that the shape of a region where the elastic tensor differs from a known background elastic
tensor, can be detected by a simple monotonicity relation. In addition, we apply a linearization method in orderto improve the qualitiy of the shape reconstruction and provide an insight into the numerics.

162

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MODEL AND SOURCE IDENTIFICATION PROBLEMS FOR A SYSTEM OFADVECTION-REACTION EQUATIONS AND APPLICATIONS IN WATER QUALITY
Thành Trung Nguyen
Department of Mathematics, Rowan University,201 Mullica Hill Road, Glassboro, NJ 08012, USA,nguyent@rowan.edu

Abstract
We discuss coefficient and source identification problems for a system of one-dimensional advectionreaction equations. The considered equations describe the transportation of pollutants in rivers or streams. We
will discuss the stability using Carleman estimates and numerical methods for solving these inverse problems.
We will demonstrate the performance of the proposed algorithms using simulated data generated using a realistic
scenario.
This is a joint work with Dinh Nho Hao, Nguyen Van Duc, and Nguyen Van Thang.
The research of this talk is supported by Vingroup Innovative Foundation under grant number
VINIF.2020.DA16.

163

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ELECTRICAL IMPEDANCE TOMOGRAPHY WITH PARTIAL CAUCHY DATA
Tran Nhan Tam Quyen
Institute for Numerical and Applied Mathematics, University of Goettingen, Germany.

Abstract
In this presentation we would like to present the problem of identifying the conductivity and/or thereaction coefficient in elliptic PDEs from several sets of Cauchy data on an accessible part of the boundary. The
variational method of the finite element discretization combining with the regularization technique is applied totackle the ill-posed identification problem.
The stability of the proposed approach and the convergence of the finite element regularization
approximations to the sought coefficients are discussed, which confirm that the coefficients distributed inside thephysical domain can be reconstructed from a finite number of partial Cauchy data.

164

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A REDUCED BASIS ENSEMBLE KALMAN METHOD FOR INVERSE PROBLEMS
F. Silvaa, C. Pagliantinia, M. Greplb, K. Veroyaa
CASA, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands, f.a.b.silva@tue.nl;b
IGPM, RWTH Aachen University, 2056, Aachen, Germany, grepl@igpm.rwth-aachen.de;aAbstract
In the process of reproducing the state dynamics of distributed-parameter systems, data from physicalmeasurements can be incorporated into the mathematical model to reduce the parameter uncertainty and,
consequently, improve the state prediction. This process, referred to as Data Assimilation, must deal with thedata and model misfit arising from experimental noise as well as model inaccuracies and uncertainties. In ourstudy, we focus on the ensemble Kalman method (EnKM) [1], an iterative Monte Carlo method indicated for thea posteriori analysis of time series. The method is gradient free and, just like the ensemble Kalman filter, relies
on a sample of parameters or particles ensemble to identify the state that better reproduces the physicalobservations, while preserving the physics of the system as described by the model.
In this context, for physical problems described by non-linear parabolic partial differential equations,
we employ reduced order modeling (ROM) techniques [2] to generate surrogate models of different accuracy.
These are combined with the EnKM to study its behavior in the presence of increasing levels of experimental
noise and for model errors of different magnitude. We also experimentally investigate the role of the ensemble
size on the reconstruction error and extend the method by implementing a poll-voting feature for improved
parameter estimation. We emphasize that such experiments, involving unknown distributed parameters in two or
more spatial dimensions, are potentially very expensive and are made possible by the remarkable efficiency
provided by the surrogate models.

References
1. M. Iglesias, K. Law and A. Stuart, Ensemble Kalman Methods for Inverse Problems, Inverse Problems, 29,(2013).
2. N. Aretz, F. Silva, M. Grepl and K. Veroy, Space-Time Formulation for Parabolic Problem: a Certified
Reduced Basis Approach, in preparation.

165

MINISYMPOSIUM
M18: Coefficient Identification Problems
Organizer:
Daniel Lesnic, University of Leeds, UK, D.Lesnic@leeds.ac.uk
Coefficient identification problems occur naturally in almost any mathematical modelling ofphysical phenomena. They are ill-posed and in general nonlinear. Various aspects concerning theexistence and uniqueness of solution are delicate to answer. Furthermore, the stability issue and thenumerical reconstruction methods are of interest.
We want to bring together experts and young researches working in this field to discuss about
new results in the analysis and numerics of coefficient identification problems.

166

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AN INVERSE PROBLEM FOR THE STURM-LIOUVILLE EQUATION WITHANALYTICAL DEPENDENCE ON THE EIGENPARAMETER IN A BOUNDARYCONDITION

N. Bondarenkoa,baDepartment of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, 410012 Saratov,
Russia, bondarenkonp@info.sgu.ru;b Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse
34, 443086 Samara, Russia

Abstract
The talk concerns the following eigenvalue problem: y' ' ( x)  q( x) y( x)  y( x), x  (0,  ),y(0)  0,(1)

f1 ( ) y' ( )  f 2 ( ) y( )  0.

(2)
Here (1) is the standard Shurm-Liouville equation with the complex-valued potential q from L2(0,π), but theboundary conditions (2) contain arbitrary functions f1 and f2 entire by the spectral parameter. We consider the
inverse spectral problem, which consists in recovering the potential q from a part of the spectrum {n }n1 ,satisfying some conditions. The functions f1 and f2 are supposed to be known a priori.
Such inverse problem statement generalizes various inverse spectral problems applied in science andengineering: The Hochstadt-Lieberman problem [1];
 The inverse transmission eigenvalue problem [2]; Partial inverse problems for Sturm-Liouville operators with discontinuities [3]; Partial inverse problems for quantum graphs [4].
We will discuss the following issues of our inverse problem:Uniqueness;
Constructive algorithm for solution;
Sufficient conditions of solvability;
Local solvability and stability.
Our approach is based on the methods of [5]. We reduce the considered inverse problem to the classicalSturm-Liouville inverse problem by two spectra, by using a special Riesz basis of vector-functions.
The work is supported by Grant 20-31-70005 of the Russian Foundation for Basic Research.

References
1. H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl.
Math., 34, 676-680 (1978).
2. J.R. McLaughlin and P.L. Polyakov, On the uniqueness of a spherically symmetric speed of sound fromtransmission eigenvalues, J. Diff. Eqns., 107, 351-382 (1994).
3. O. Hald, Discontinuous inverse eigenvalue problem, Commun. Pure Appl. Math., 37, 539-577 (1984).
4. N.P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, Anal.
Math. Phys., 8, 155-168 (2018).
5. C.-F. Yang, N.P. Bondarenko and X.-C. Xu, An inverse problem for the Sturm-Liouville pencil with arbitrary
entire functions in the boundary condition, Inv. Probl. Imag., 14, 153-169 (2020).

167

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE PROBLEMS TO RECONSTRUCT SPACE- AND TIME-DEPENDENTSOURCES IN EVOLUTION EQUATIONS CONTAINING FRACTIONAL LAPLACEOPERATORSNataliia Kinash and Jaan Janno
Tallinn University of Technology Ehitajate tee 5, 19086 Tallinn, Estonia
nataliia.kinash@taltech.ee

Abstract
In the paper [1] we considered an inverse source problem for the equation

D0 u ( x, t )  u ( x, t )  F ( x, t ),x    Rn ,t  (0, T ) ,where D0 is Caputo or Riemann-Liouville fractional time derivative of the order   R  \ N with lower

terminal 0 and  the Laplacian. We assumed that t  (0, T ) and supp F    (0, t0 ) and posed a
problem to determine u ( 0 ,t ) and F ( 0 ,t ) on the basis of given u ( t ,T ) . We proved uniqueness of a000solution of this problem and deduced solution formulas. The non-locality of the time derivative D0 was animportant prerequisite for achieving these results.
In this talk we consider an analogous problem in space directions. We follow the equation

D  u( x, t )  () u( x, t )  F ( x, t ), x  R n , t  (0, T )where D  is a time derivative of integer or non-integer order,  () is the fractional Laplacian of theorder   (0,1) given by(   ) v ( x )   1 

2 v ( ),and  is the Fourier transform, i.e.

v( )   n exp( i  x)v( x)dx
R

We assume that  is an open subset of R n and supp F  ( R n \  )  (0, T ) and formulate the following
inverse problem: given u  ( 0 ,T ) find u ( R n  )( 0 ,T ) and F ( R n  )( 0 ,T ) .

Reference
1. N. Kinash, J. Janno An Inverse Problem for a Generalized Fractional Derivative with an Application in
Reconstruction of Time- and Space- Dependent Sources in Fractional Di_usion and Wave Equations, Mathematics 7(2019), 1138.

168

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECOVERY OF A SPACE-DEPENDENT RATE OF REACTION IN A THERMALWAVE MODEL OF BIO-HEAT TRANSFER

D. Lesnic and M. Alosaimi
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK, D.Lesnic@leeds.ac.uk

Abstract
This work investigates the numerical retrieval of the space-dependent blood perfusion coefficient (rateof reaction) in a newly derived dimensionless thermal-wave model of bio-heat transfer. The additional
measurement for such reconstruction is the upper-base temperature or a pair of partial Cauchy data. The
considered inverse problem is reformulated as a non-linear least-squares minimization problem. This
minimization problem is solved by a minimization procedure based on the MATLAB routine lsqnonlin in
conjunction with a direct solver of unconditionally stable finite difference method.
Several numerical tests for verifying the convergence and stability of the proposed algorithm areillustrated and discussed.

169

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

IDENTIFICATION OF HETEROGENEOUS MATERIAL COEFFICIENTS USINGULTRASONIC ARRAYS
Anthony J. Mulhollanda , Katy M.M. Tantb , David P. Bournec, Russell Nivenb and Smita Sahuba

Department of Engineering Mathematics, University of Bristol, Bristol, UK, BS8 1UB.
anthony.mulholland@bristol.ac.ukbDepartment of Mathematics and Statistics, University of Strathclyde, Glasgow, UK, G1 1XH.
katy.tant@strath.ac.uk, russell.niven@strath.ac.uk
cDepartment of Mathematics, Heriot-Watt University, Edinburgh, UK, EH14 4AS d.bourne@hw.ac.ukdSchool of Mathematics and Physics, University of Portsmouth, UK, PO1 2EG. smita.sahu@port.ac.uk

Abstract
Ultrasonic phased arrays are used for non-destructive imaging of safety critical structures such as thosefound in nuclear plants. Flaw imaging methods normally assume that the host material is a homogeneous solid
and this can lead to reduced flaw detection and characterisation when the host material has heterogeneousmaterial coefficients. Knowledge of the spatial variation in the material’s internal elastic properties can therefore
be used to improve flaw imaging. This paper will discuss techniques to reconstruct the spatially varying
coefficients (the local orientation of polycrystalline materials) from ultrasonic phased array data. A Voronoi
tessellation is used to parameterise the material's structure, where the coefficient of interest is constant in eachcell. In the first instance, the reversible-jump Markov Chain Monte Carlo (rj-MCMC) method is used to sample
the posterior distribution of this piecewise constant heterogeneous material.
In the second instance we solve this non-smooth, non-convex optimisation problem using a multi-start
nonlinear least squares method on a square grid. Good reconstructions are achieved but the method is shown tobe sensitive to the addition of noise. We prove that the orientations can be determined uniquely given enoughboundary measurements and provide a numerical method that is relatively stable with respect to the addition ofnoise. The reconstructed material map is then used with an adapted flaw imaging algorithm and empirical results
suggest that this produces more reliable reconstruction of defects.

170

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AN ITERATION METHOD FOR SOLVING THE INVERSE PROBLEM OF
FREEZING SOIL
B. Rysbaiulya and N. RysbayevabaInternational Information Technology University, Almaty, Kazakhstan, b.rysbaiuly@mail.ru;bInstitute of Mechanics and Engineering, University of Bordeaux, Bordeaux, France, n.rysbayeva@mail.ru

Abstract
At conductive heat transfer, the heat flow q is proportional to the temperature gradient q  kT ,where k – heat conductivity coefficient, T – temperature [1]. During the freezing process of soil, the following
three zones are formed: thawed, phased and frozen zones. The fundamental law of conductive mechanism for heattransfer is valid for phased zone transitions. This leads to the conclusion that the heat conductivity coefficientk (T ) will change along with temperature. Assume that k (T ) – heat conductivity coefficients of thawed, phased
and frozen soil; Ws – quantity of firmly bound water in soil; W – quantity of unfrozen water;  p – isotherm ofphased zone;  – isotherm of frozen zone. Moreover, hb (t ) – boundary between thawed and phased zones, andh(t ) – boundary between phased and frozen zones. c(T ) – coefficient of volumetric heat capacity. We believethat at moving boundaries hb (t ) and h(t ) conditions of temperature continuity are satisfied [2]:T hb (t )  0, t   T hb (t )  0, t    p ,T h(t )  0, t   T h(t )  0, t    ,where t  0 – time. By
considering the features of physical process of moisture freezing in soil, we accept boundary conditions of 4th kindk  / zh (t )   pdhb (t ) / dt, k  / zh(t )   0 . We also use the measured value of soil temperatureb

at the Earth’ surface. It is required to find such value of parameter k(T) that gives a minimum to the functional J(k).
In this presentation, an iterative method for calculating the coefficient k (T ) is developed. Moreover,based on experimental research done by scientists, a nonlinear model of temperature transfer in soil is built;
An iterative formula for calculating heat conductivity coefficient k (T ) is developed;
Limitation of approximate value of thermal conductivity coefficient k (T ) is proved, and the monotonyof minimized functional J (T ) is verified;Nonlinear difference problem is solved by the Newton method and method of choosing the initialapproximation of the iterative process is indicated;
Quadratic convergence of the Newton method is proved;
Software program was written, and numerical calculations were conducted;
Analysis of convergence of the Newton method and assessment of the accuracy of the developed method
were processed.

References
1. J. Berger, D. Dutykh, N. Mendes and B. Rysbaiuly. A new model for simulating heat, air and moisture transportin porous building materials, International Journal of Heat and Mass Transfer, Volume 134, 1041–1060 (2019).
2. B. Rysbaiuly, M. Ryskeldi, A. Kulzhanov and K. Rysbayeva. Inverse problems of heat and mass transfer in one
layer and multilayer walling, Global Journal of Pure and Applied Mathematics, 11 (2018).
3. B. Rysbaiuly, N. Yunicheva and N. Rysbayeva. An Iterative Method to Calculate the Thermal Characteristics of
the Rock Mass with Inaccurate Initial Data. Journal of Open Engineering, Scopus, 2016.

171

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

UNIQUENESS FOR AN INVERSE SOURCE PROBLEM OF DETERMINING A
SPACE DEPENDENT SOURCE IN A NON-AUTONOMOUS PARABOLIC
EQUATIONMarian Slodicka
Department of Electronics and Information Systems, Ghent University, Krijgslaan 281, S8, Ghent,
Belgium, marian.slodicka@ugent.be

Abstract
We consider a bounded domain   R N , N  1 with su_ciently smooth boundary  . We areinterested in the uniqueness of determining of an unknown couple u ( x, t ), f ( x)  obeying the following
linear non-autonomous parabolic problem of second orderu ( x, t )  A(t )u ( x, t )  h(t ) f ( x)  F ( x, t ), in   (0, T ),u ( x, t )  g ( x, t ), on   (0, T ),u ( x,0)  u0 ( x), in ,


(1)

where

u ( x, T )  uT ( x) .

(2)

Most of the known results deal with recovery for autonomous operator A , e.g. [1]. Isakov [2, Theorem
2.1] presented uniqueness result for A(t ) assuming some monotonic in time behavior of coeffcients at

A(t ) and 0  h(t ) , 0   t h . Slodicka and Johansson [3] also needed monotone in time assumption forA(t ) andh' (t ).
0  h ,  ' (t )  0 , where  (t ) :h(t )The aim of this talk is to present some uniqueness results for the ISP just for 0  h in the nonautonomous case A(t ) .
References
1. A.I. Prilepko, D.G. Orlovsky, and I.A. Vasin. Methods for Solving Inverse Problems in Mathematical Physics.
Pure and Applied Mathematics, Marcel Dekker. 231. New York, NY: Marcel Dekker, 2000.
2. Victor Isakov. Inverse parabolic problems with the final overdetermination. Communications on Pure andApplied Mathematics, 44(2):185{209, 1991.
3. M. Slodicka and B. T. Johansson. Uniqueness and counterexamples in some inverse source problems. Applied
Mathematics Letters, 58:56{61, 2016.

172

MINISYMPOSIUM
M19: Modern Challenges in Imaging, Tomography, and Radon Transforms
Minisymposium dedicated to the 70th anniversary of an outstanding expert in inverse problems
and imaging sciences, Robinson Professor of Mathematics Tufts University, Todd Quinto
Organizers:
Jan Boman, Stockholm University, Stockholm, jabo@math.su.se Stockholm, jabo@math.su.se
Ming Jiang, Peking University, China, ming-jiang@pku.edu.cn (ask to remove, will not come)Venky Krishnan, Tata Institute of Fundamental Research (TIFR), India, vkrishnan@tifrbng.res.inRoman Novikov, Ecole Polytechnique, France, roman.novikov@polytechnique.edu
Imaging, Tomography and Radon Transforms are one of the fastest developing areas ofinverse problems, and they have a wide range of applications. Tomography became generally known
in the seventies when medical diagnosis was revolutionized by X-ray computed tomography. The
mathematical model is an integral transform and Fourier integral operator--the Radon transform. Other
imaging technologies lead to more complicated integral transforms as well as linear and nonlinear
problems. Classical problems, such as limited-data tomography, are still challenging. Cone-Beamtomography, ultrasound tomography, multi-modal imaging, Compton and Bragg tomography, andtime-dependent problems are other frontiers of research. All these technologies have in common thatthe mathematical models are ill-posed inverse problems.
The minisymposium celebrates one of the pioneers of the mathematics of imaging andtomography, Professor Eric Todd Quinto, on the occasion of his 70th birthday. It will bring together
well-established scientists and young researchers to introduce new methods in the field, including themethods pioneered by Professor Quinto.

173

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AN ADAPTIVE NONPARAMETRIC ESTIMATOR IN AN INVERSE PROBLEM FOR
EXPONENTIAL RADON TRANSFORM

Anuj Abhishek
Department of Mathematics,University of North Carolina, Charlotte, USA,anuj.abhishek@uncc.edu

Abstract
In this work, we propose a locally adaptive strategy for estimating a function from its Exponential
Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated.
We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide
Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a log n factor. We also
show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used
and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.
This is joint work with Dr. Sakshi Arya, Department of Statistics, Pennsylvania State University.

174The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DOMAINS WITH ALGEBRAIC X-RAY TRANSFORM
Mark Agranovsky
Bar-Ilan University, Israel
agranovs@math.biu.ac.il

Abstract
Koldobsky, Merkurjev and Yaskin proved that given a convex body K in R^n, n is odd, with smoothboundary, such that the volume of the intersection of K with a hyperplane L (the sectional volume function)
depends polynomially on the distance of L to the origin, then the boundary of K is an ellipsoid. In evendimension, the sectional volume functions are never polynomials, nevertheless in the case of ellipsoids theirsquares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity ofthe dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions
are roots (of some power) of polynomials and confirm this conjecture for planar domains, bounded by algebraic
curves. A multidimensional version in terms of chords lengths, i.e., of the X-ray transform of the characteristic
function, is given.
The result is motivated by Arnold's conjecture on characterization of algebraically integrable bodies.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MICROLOCAL ANALYSIS OF THE CROSSWELL AND THE WALKAWAYSEISMIC DATA

Raluca Felea
Rochester Institute of Technology, School of Math Sciences, Rochester, NY, 14623, USA
rxfsma@rit.edu

Abstract
We study the microlocal properties of the forward operator, F (a Fourier integral operator), which maps
the image to the data, in two special cases of the the borehole seismic data: crosswell and the walkaway
geometry. In general, in the borehole seismic acquisition data, the receivers are located on a vertical borehole
and the sources are in another borehole or on the surface. The relation between the singularities of the image andthe singularities of the data is given by the canonical relation, C, of the forward operator, F. To understand theoperator F, we consider the projections to the left and to the right of the canonical relation, C. We are alsointerested in the artifacts which appear by applying the backprojection operator, F*, to the forward operator (tofind the image).
In the crosswell geometry, the sources are located in another vertical borehole, and in the walkaway
geometry, the sources are on a line on the surface, passing over the borehole top. In both cases, we show thatthe canonical relation is singular at the union of two smooth hypersurfaces which intersect transversally. The
singularities of the right and the left projections of the canonical relation, are folds and blowdowns. Also, in
these two cases, we conclude that the artifacts are very strong.
This is joint work with A. Greenleaf, R. Gaburro and C. Nolan.

176

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MICROLOCAL ANALYSIS IN TOMOGRAPHIC RECONSTRUCTIONSJürgen Frikel
Department of Computer Science and Mathematics, Ostbayerische Technische Hochschule Regensburg,
Prüfeninger Straße 58, 93049 Regensburg, Germany
juergen-frikel@oth-regensburg.de

Abstract
During recent years, microloacal analysis has established itself as one of the standard tools for analyzing
tomographic reconstructions. It has been shown that microlocal anaylsis is particularly useful for analyzing andunderstanding reconstrctions from incomplete data. In particular, using microlocal analysis one can gain valuable
insights that allow for to a better interpreation of reconstruted images and that can be used to design dedicatedalgorithms for incomplete data problems. Specifically, microlocal analysis allows to characterize image featuresthat can be realiably reconstructed from available tomographic data, and to explain why singular artifacts are
generated in several classical imaging scenarios. Also, it provide strategies that allow to reduce or even avoid thegeneration of artifacts.
In this talk I will explain the basic concepts of microlocal analysis and review a framework foranalyzing reconstructions in incomplete data tomography. In particular, I will discuss fundamental results thatcharacterize visible singularities and singular artifacts. I will also outline how those results have contributed tothe practical development of algorithm in different applications of tomography.
This talk is dedicated to Eric Todd Quinto in honor of his 70 th birthday, who has made major
contributions to both fields, mathematics of tomographic reconstruction and microlocal analysis.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RESOLUTION OF 2D RECONSTRUCTION OF FUNCTIONS WITH NONSMOOTH
EDGES FROM DISCRETE RADON TRANSFORM DATA

Alexander Katsevich
University of Central Florida, USAe-mail: alexander.katsevich@ucf.edu

Abstract
Let be an unknown function in, andbe its reconstruction from discrete Radon transform data,
where is the data sampling rate. We study the resolution of reconstruction when has a jump discontinuity
along a nonsmooth curve . The assumptions are that (a)is an-size perturbation of a smooth curve ,and (b)is Holder continuous with some exponent. We compute the Discrete Transition Behavior(or, DTB) defined as the limit
whereis generic. We illustrate the DTB by
two sets of numerical experiments. In the first set, the perturbation is a smooth, rapidly oscillating sinusoid, andin the second - a fractal curve. The experiments reveal that the match between the DTB and reconstruction isworse asgets more rough. This is in agreement with the proof of the DTB, which suggests that the rate ofconvergence to the limit is. We then propose a new DTB, which exhibits an excellent agreement withreconstructions. Investigation of this phenomenon requires computing the rate of convergence for the new DTB.
This, in turn, requires completely new approaches. We obtain a partial result along these lines and formulate aconjecture that the rate of convergence of the new DTB is.

178

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARIZATION BY ARCHITECTURE: LEARNING WITH FEW DATA AND
APPLICATIONS TO CT
Peter Maas
University of Bremen, Germany
e-mail: pmaass@uni-bremen.de

Abstract
We start with a basic introduction on deep learning approaches to inverse problems. We then focus on
the learned ISTA concept and describe it as a method for learning a data dependent optimized Tikhonovfunctional. The main part of the talk is on learning with few data. In particular we investigate deep prior
networks for solving inverse problems. Using the LISTA architecture in a deep prior network allows to proof
equivalences to classical regularization schemes. On the experimental side we focus on low dose CT
reconstructions.
We present a standardized data set and perform a numerical comparison of different deep learning
concepts. The comparison is in terms of accuracy but also in terms of the amount of test data needed for training.
We close the talk with an overview of our plan for future research (DL for parametric PDEs, magnetic particle
imaging).
This is the joint work with Johannes Leuschner, Maximilian Schmidt, Sören Dittmer, Daniel Otero
Baguer, Clemens Arndt.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARIZED RECYCLING METHODS FOR LINEAR INVERSE PROBLEMSWITH APPLICATIONS TO ADAPTIVE OPTICS
Ronny Ramlau and Kirk Soodhalter
Industrial Mathematics Institute Johannes Kepler University, Austria

Abstract
Subspace recycling techniques have been used quite successfully for the acceleration of iterative
methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace
generated iteratively by a known algorithm with a fixed subspace of vectors which are ``useful' for solving the
problem. Often, this has the effect of inducing a projected version of the original linear system to which the
known iterative method is then applied, and this projection can act as a deflation preconditioner, accelerating
convergence.
In this talk we consider subspace augmentation-type iterative schemes applied to a linear ill-posed
problems in a continuous Hilbert space setting, based on a recently developed framework describing thesemethods. We show that under suitable assumptions, a recycling method satisfies the formal definition of aregularization, as long as the underlying scheme is itself a regularization. We then develop an augmented
subspace version of the gradient descent method and demonstrate its effectiveness on problems arising in
Adaptive Optics for the resolution of large sky images by ground-based extremely large telescopes.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

QUANTITATIVE INVERSE PROBLEMS IN VISCO-ACOUSTIC MEDIA ANDEVALUATION OF ATTENUATION MODEL UNCERTAINTIES
Otmar Scherzer
University of Vienna, Austria, otmar.scherzer@univie.ac.at

Abstract
We consider the inverse problem for the quantitative reconstruction of physical parameters in viscoacoustic media, for the recovery of tissue features and identification of embedded bodies. Attenuation is encodedin wave equations. There exists a zoo of such methods in the literature, which are reviewed in the course of thistalk.
Finally, we review inverse problems of reconstruction of parameters in visco-acoustic media. Particularemphasis will be given to study a mismatch of models used in forward and inverse simulations.
This is joint work with Florian Faucher, Inria Bordeaux Sud-Ouest, Université de Pau et des Pays de

181

MINISYMPOSIUM
M20: Inverse Obstacle and Control Problems in Mechanics
Organizers:
Hiromichi Itou, Tokyo University of Science, Japan, h-itou@rs.tus.ac.jp
Victor A. Kovtunenko, University of Graz, Austria, and Lavrentyev Institute of Hydrodynamics SB RAS,
Novosibirsk, Russia, victor.kovtunenko@uni-graz.at
Gennadii V. Alekseev, Far East Federal University, and Institute of Applied Mathematics FEB RAS,
Vladivostok, Russia, alekseev@iam.dvo.ru
Mikhail M. Lavrentiev, Jr., Novosibirsk State University, Novosibirsk, Russia, mmlavrentiev@gmail.com
Variational approaches are widely used in all fields of inverse and related ill-posed problems,
nonsmooth optimization, optimal control, and homogenization. In this respect, we focus but are not limited
to singular and unilaterally constrained problems arising in mechanics, geophysics, and real-world kinetics
which are governed by complex systems of PDE equations and inequalities. In a broad scope, theminisymposium objectives are directed toward advances which are attained in the theory, computation, andapplication of inverse and ill-posed problems.

182

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A NUMERICAL METHOD FOR SOURCE IDENTIFICATION PROBLEM RELATEDTO DYNAMICAL KIRCHOFF PLATE EQUATION

O. Baysal
Department of Mathematics, University of Malta, Malta.
e-mail: onur.baysal@um.edu.mt

Abstract
Kirchoff Plate model is an integral part of most engineering fields including plate and shell structures[1]. In [2], some important identification problems are stated and some properties are analyzed such as stability
and uniqueness. In this work we study the inverse problem of identifying the unknown load distribution
inthe rectangular domainsuch thatHereis the displacement at a point
normal derivative of ,is the (known) temporal load,
is the Poisson’s ratio, is the elasticity modulus andand a time,denotes theis the bending stiffness,where

In determination of we have the following boundary observation on
The conjugate gradient algorithm (CGA) is designed for the numerical solution of the identification
problem. The proposed approach is based on weak solution theory for PDEs, Tikhonov regularization combinedwith the adjoint method. Computational results, obtained for noisy output data, are illustrated to show anefficiency and accuracy of the proposed approach, for typical classes of source functions.

References
1. L. Fryba, Vibrations of the Solids and Structures under Moving Loads, Thomas Telford Publishing House,
1999.
2. M. Yamamoto, Determination of forces in vibrations of beams and plates by point wise and line observations,
J. Inv. Ill-Posed Problems, Vol.4, No.5, pp.437-457 1996.

183

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

HIGH RESOLUTION INVERSE OBSTACLE SCATTERING USING MULTIPLE
FREQUENCY DATA
C. Borgesa , L. Greengardb,c and A. GillmandaDepartment of Mathematics, UCF, Orlando, Florida, USA, e-mail: carlos.borges@ucf.edu
bCIMS - NYU, New York, New York, USA, greengar@cims.nyu.edu
cFlatiron Institute, New York, New York, USA, lgreengard@flatironinstitute.orgdDepartment of Applied Mathematics, UC - Boulder, Boulder, Colorado, USA, adrianna.gillman@colorado.edu

Abstract
The problem of recovering the sound profile of a penetrable medium represented by a compactly
supported function q(x) given measurements of the far field generated by the scattering of plane waves frommultiple known angles of incidence and multiple known frequencies can be cast as

F (q)( x)  u scat ( x)with F and u

scat(x) denoting the forward scattering operator and scattered field, respectively.
scat
To calculate the value of the scattered field u
off of q(x) given an incident plane wave uinc
scat
ikxd
2
for x  R one must solve the variable-coefficient Helmholtz equation
u
e
scat
u
( x)  k 2 (1  q( x))u scat ( x)  k 2 q( x)u inc ( x) in x  R 2 \ D ,together with the Sommerfeld radiation condition. We use the HPS fast direct solver for the Helmholtz equation

[4]. The computational cost of this solver is O N

3/ 2

 for the factorization step and ON  for each newright-hand-side, where N is the number of points used to discretize the domain.
~ of the unknown contrast continuous function
The inverse problem of finding an approximation q

q(x) given measurements of the far field u scat scattered by a collection of incident plane waves, can becasted as the nonlinear optimization problem~  argmin u scat ( x )  F ( q ) .
q(1)

We provide a fast, stable algorithm for the solution of this problem that consists of higher-dimensional
extensions of the techniques used in [1]. Problem (1) is both nonlinear and ill-posed. We address nonlinearity
through the use of Gauss-Newton’s iteration for the frequency, employing the Fréchet derivative of F withrespect to q . Despite the fact that the underlying optimization problem is formally ill posed and non-convex,
our technique requires only the solution of a sequence of linear least-squares problems at successively higher
frequencies, known as the recursive linearization algorithm [3]. By seeking a suitably band-limited
approximation of the sound speed profile, each least-squares calculation is well conditioned and involves thesolution of a large number of increasingly computationally complex direct scattering problems [2].

References
1. C. Borges and L. Greengard, Inverse obstacle scattering in two dimensions with multiple frequency data andmultiple angles of incidence, SIAM J. Imaging Sciences, 8 (2015), pp. 280–298.
2. C. Borges, L. Greengard, A. Gillman, High resolution inverse scattering in two dimensions using recursive
linearization SIAM J. Imaging Sciences, 10 (2017), pp. 641–664
3. Y. Chen, Inverse scattering via Heisenberg’s uncertainty principle, Inverse Problems, 13 (1997), pp. 253-282.
4. A. Gillman, A. Barnett, P. Martinsson, A spectrally accurate direct solution technique for frequency-domain
scattering problems with variable media, BIT. 55 (2014), pp. 141–170.

184

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

AN INVERSE OBSTACLE PROBLEM FOR THE TIME-DEPENDENT HEATEQUATION

Peter Elbau
University of Vienna, Austria
peter.elbau@univie.ac.at

Abstract
In this work we deal with the inverse problem of reconstructing the interior boundary curve of a doublyconnected bounded domain given the temperature and the thermal flux on the known exterior boundary curve.
We reduce the time-dependent problem to a sequence of stationary problems by applying the Laguerre transformwith respect to the time variable. We represent the solution as a modified single-layer potential whose densities
have to satisfy a sequence of systems of non-linear boundary integral equations. An iterative algorithm is derived
for the numerical solution of the system. We linearize the system using the Fréchet derivative of thecorresponding integral operator and we apply Tikhonov regularization to handle its ill-posedness. The numericalresults show the feasibility of the proposed scheme.
This is a joint work with R. Chapko (Ivan Franko National University of Lviv, Ukraine).

185

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

IMPLEMENTATION OF THE ENCLOSURE METHOD FOR SOME INVERSECRACK PROBLEMS

Andreas S. Hauptmann
University College London, UK;University of Oulu, Finlanda.hauptmann@ucl.ac.uk

Abstract
An algorithm is introduced for using electrical surface measurements to detect and monitor cracks
inside a two-dimensional conductive body. The technique is based on transforming the probing functions of theclassical enclosure method by the Kelvin transform. The transform makes it possible to use virtual discs forprobing the interior of the body using electric measurements performed on a flat surface.
Theoretical results are presented to enable probing of the full domain to create a profile indicatingcracks in the domain.
Feasibility of the method is demonstrated with a simulated model of attaching metal sheets together by
resistance spot welding.
This is joint work with Masaru Ikehata, Hiromichi Itou, and Samuli Siltanen

186

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

UNIQUENESS IN LOAD IDENTIFICATION IN VIBRATING NANOPLATES

Alexandre Kawano
Escola Politecnica da Universidade de Sao Paulo, PMR, Av. Prof. Mello Moraes 2231,
Sao Paulo, 05508-900 Brazil
akawano@usp.br

Abstract
Nanoplates present numerous applications in micro and nano-electromechanical systems as sensors for
mass, energy harvesters, or for vibration control. Recently, applications such as the measurement of forces
exerted by cells or the intraocular pressure have been proposed in the literature. As far as we know, it is the firsttime that the uniqueness in the inverse problem m identifying pressure loads acting over a nanoplate from themeasurement of its dynamic response is treated. We considered the general case in which a finite number ofsources is present, that is, for loads of the typefunctionsg m (t )

M
are
m1



M

m 1

g m (t ) f m (t ) , M  1 , where the time-dependentassumed to be known and linearly independent on any sub-interval of theregistration time interval. We prove that if the transverse displacement of the nanoplate is known in an open
subset compactly contained in the plate domain for an arbitrarily small time interval, then it is possible toidentify uniquely theloads  f m ( x)m1 . Furthermore, we proposed a new method that retakes the idea of using spherical means
Mthat can be used in other contexts.
The presentation may interest Engineers, Mathematicians and Physicists working on dynamical
problems involving nanostructures and related inverse problems. It is based on the work carried out inconjunction with Antonino Morassi and Ramón Zaera [1].

References
1. Alexandre Kawano, Antonino Morassi, and Ramón Zaera. Inverse load identification in vibrating nanoplates.
Submitted, 2021.

187

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MATHEMATICAL MODEL OF CRACK DIAGNOSISVictor A. Kovtunenkoa,ba Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz,
Heinrichstr.36, 8010 Graz, Austria, victor.kovtunenko@uni-graz.at;b Lavrent'ev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090
Novosibirsk, Russia

Abstract
In this contribution, an optimization technique for identification of single defects like cracks [3] ispresented. The respective inverse problem is aimed at crack diagnosis motivated by applications to nondestructive testing with acoustic, electromagnetic, and elastic waves in engineering sciences.
The inverse acoustic scattering model for crack diagnosis is described by Helmholtz problem within mathematic
framework and investigated for the sake of scientific computing. Minimizing the misfit from given
measurements leads to an optimality condition based imaging function. It is used for non-iterative identificationof the center of an unknown crack put in a test domain and characterized by complex-valued surface impedanceas suggested in [5]. Further the approach is extended in [1] to the Lippmann-Schwinger equation describing
small inhomogeneities by the mean of complex-valued refractive index. In [4] the numerical tests are presented
for a cluster of cracks of T-junction shape and are carried out based on the Petrov-Galerkin generalized FEM
using local wavelets and level-sets [6]. This shows high-precision identification result and stability to noisy dataof the diagnosis, which is illustrated for sound-soft as well as moderately sound-hard cracks when varying thecoefficient of surface impedance.
The shape optimization approach is developed in [7] for overdetermined and state-constrained optimization
problems of tracking type, and in [2] for identification of breaking lines (a free discontinuity allowing jumps).
The corresponding asymptotic analysis realizes on methods of regular as well as singular perturbations.
Acknowledgment. The author thanks the Russian Foundation for Basic Research (RFBR) joint with JSPS
research project 19-51-50004 for partial support.

References
1. F. Cakoni, V.A. Kovtunenko, Topological optimality condition for the identification of the center of an
inhomogeneity, Inverse Probl., 34, 035009 (2018).
2. D. Ghilli, V.A. Kovtunenko, K. Kunisch, Inverse problem of breaking line identification by shape
optimization. J. Inverse Ill-posed Probl., to appear.
3. A.M. Khludnev, V.A. Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000.
4. V.A. Kovtunenko, Mathematical model of crack diagnosis: inverse acoustic scattering problem and its highprecision numerical solution, Vibroengineering PROCEDIA, 22, 31-35 (2019).
5. V.A. Kovtunenko, K. Kunisch, High precision identification of an object: optimality conditions based conceptof imaging, SIAM J. Control Optim., 52, 773-796 (2014).
6. V.A. Kovtunenko, K. Kunisch, Revisiting generalized FEM: a Petrov-Galerkin enrichment based FEM
interpolation for Helmholtz problem, CALCOLO., 55, 38 (2018).
7. V.A. Kovtunenko, K. Ohtsuka, Shape differentiability of Lagrangians and application to overdetermined
problems, In: H. Itou, S. Hirano, M. Kimura, V.A. Kovtunenko, A.M. Khludnev (eds.) Mathematical Analysis
of Continuum Mechanics and Industrial Applications III (Proc. CoMFoS18), Ser. Mathematics for Industry,
Springer, Singapur, to appear.

188

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON HOMOGENIZATION OF PERIODIC HYPERBOLIC SYSTEMS

Yulia Meshkova
Chebyshev Laboratory, St. Petersburg State University
juliavmeshke@yandex.ru

Abstract
This talk is devoted to homogenization of solutions of periodic hyperbolic systems with rapidly
oscillating coefficients. Classical results in homogenization theory look as the convergence of solutions of the
problem with rapidly oscillating coefficients to the solution of the so-called effective problem with constantcoefficients. The constants in the corresponding error estimates depend on the differential operator, the lattice of
periodicity, and the initial data somehow.
We are interested in the operator error estimates. In such estimates, dependence on the initial data in theerror estimates is explicit: we have the norm of the data in the error estimate.

189

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STEKLOV AND MODIFIED TRANSMISSION EIGENVALUES AS TARGET
SIGNATURES IN AN INVERSE FLUID-SOLID INTERACTION PROBLEM

Virginia Selgas
Department of Mathematics, University of Oviedo, EPI Gijón, 33203 Gijón, Asturias, Spain
E-mail: selgasvirginia@uniovi.es

Abstract
We investigate an inverse scattering problem for the interaction of waves in a linearly elastic solid andan acoustic fluid by using two sets of eigenvalues: Steklov eigenvalues and modified transmission eigenvalues.
For both sets of eigenvalues, we first define them for the problem at hand. We then characterize them by meansof operators suitable for the analysis of their location in the complex plane. We also discuss their approximation
using far field scattering data via suitable modified far field equations. We finally provide numerical examples
that show their approximation from synthetic far field measurements and their application as target signatures.
This is joint work with M. Levitin (University of Reading) and P. Monk (University of Delaware).

References
1. M. Levitin, P. Monk and V. Selgas, Impedance eigenvalues in linear elasticity. SIAM Journal on Applied
Mathematics, 18(6) (2021), 2433-2456.
2. P. Monk and V. Selgas, Modified transmission eigenvalues for inverse scattering in a fluid-solid interaction
problem. Research in the Mathematical Sciences, 9:3 (2022).

190

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

OPTIMIZATION APPROACH IN 2D PROBLEMS OF STATIC FIELDS CLOAKING
Yuliya E. Spivaka,baInstitute of Applied Mathematics FEB RAS, 690041, 7, Radio St., Vladivostok, Russia;b
Far Eastern Federal University, 690090, 8, Sukhanova St., Vladivostok, Russia, uliyaspivak@gmail.com

Abstract
In the last few years, devices for cloaking material objects have attracted the rapid attention in the
research fields of invisibility and metamaterials. The cloaking scheme of static magnetic fields based on thetransformation optics (TO) was proposed in [1]. This scheme using explicit solution possesses several
drawbacks. In particular, some components of tensor are singular on the inner boundary of the cloak. Further,another scheme for design of a magnetic cloaking device which is free from the above-mentioned drawback was
described in [2]. But it provides only approximate solutions which cannot be implemented because absence
required materials in the nature. One of approaches of overcoming mentioned difficulties consists of using the
optimization method for solving inverse problems [3, 4].
In this paper the results obtained by using the optimization method to solve cloaking problems with respect tostatic magnetic fields are present. The theoretical and numerical aspects of this method are discussed in two
paper’s parts, respectively.
The first part formulates cloaking problem for a 2D model of magnetic scattering. Cloaking effect isachieved by choosing functional parameters of an inhomogeneous anisotropic, in the general case, medium thatfills the cloak. Using the method proposed in [3] the problem under consideration is reduced to minimization of acertain cost functional depending on desired medium parameters. The solvability of control problem under study
is proved, the optimality system which describes the necessary conditions of extremum is derived, and based on
its analysis stability estimates of optimal solutions are established.
In the second part, an effective numerical algorithm is developed for solving problems of magnetic cloaking andshielding based on the use of the particle swarm optimization method (see details in [5]). Important properties of
optimal solutions are established. Based on these properties simple design rules are formulated allowing todesign cloaking and shielding shells, which possess the high cloaking or shielding performance in addition toeasy technical realization.
This work was supported by the Russian Foundation for Basic Research (project no 19-31-90039).

References
1. B. Wood, J.B. Pendry, Metamaterials at zero frequency, J. Phys.: Condens. Matter, 19, 076208 (2007)
2. F. Gomory, M. Solovyov et al., Experimental realization of a magnetic cloak, Science, 335, 1466-1468(2012).
3. G.V. Alekseev, Yu.E. Spivak, Theoretical Analysis of the Magnetic Cloaking Problem Based on an
Optimization Method, Differential Equations, 54, N 9, 1125-1136 (2018).
4. Yu.E. Spivak, Optimization method in 2D magnetic cloaking problems, SEMR, 16, 812-825 (2019).
5. G.V. Alekseev, D.A. Tereshko, Particle swarm optimization-based algorithms for solving inverse problems ofdesigning thermal cloaking and shielding devices, Int. J. Heat Mass Transfer, 135., 1269-1277, (2019).

191

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ELASTOPLASTIC MODELLING AND EXPERIMENTS FOR INVERSE PROBLEMSBASED ON ENTROPY APPROACHES

Ping Wu
Entropic Interface Group, Engineering Product Development, Singapore University of Technology and Design,
Singapore 487372, Singapore,
wuping@sutd.edu.sg

Abstract
I will present a novel method to inverse elastoplastic deformation issues based on a recently createdconcept termed entropy-of-resilience. The Born-Oppenheimer approximation and entropy theory were used todevelop non-linear elastoplastic equations for the measured stress and strain curve. Furthermore, by bridging thematerial's wetting entropy and the system's strain entropy, a relationship between water-contact-angle andelastoplastic strain is established, which can be used to inversely determine the stress-strain plots in elastoplastic
deformation using non-destructive water contact angle measurements. Finally, their use in inversely solving
elastoplastic problems of sand core production for metal casting and corrosion-reduced mechanical strength ofmagnesium alloys will be highlighted.
This research was supported by the Agency for Science, Technology and Research (A*STAR)(Singapore), under AME Individual Research Grant (Award No. A20E7c0108); the Ministry of Education(Singapore), under Tier 2 program (Award No. MOE2018-T2-1-163); and the National Research Foundation,
Prime Minister’s Office (Singapore), under its Marine Science Research and Development program [Award No.
MSRDP-P28].

References
1. P. Wu, B.T. Tan, J.-I. Jeong, J.-H. Yang, S. Wu, F. Anariba, Entropy of resilience: formulism and validation
using strained-induced surface wettability study on thin film alloys, J. Alloys Compd. 846 (2020) 156357.
2. B. T. Tan, P. Wu, F. Anariba, Modeling stress-strain nonlinear mechanics via entropy changes on surface
wetting using the Born-Oppenheimer approximation, Results in Engineering 13 (2022) 100349.

192

MINISYMPOSIUM
M21: Imaging Modalities: Recent Advances and Beyond
Organizer:
Cristiana Sebu, University of Malta, Malta, cristiana.sebu@um.edu.mt
The past century has witnessed accelerated development in imaging techniques for medical,
biological, industrial and geophysical applications using a wide range of physical modalities.
Examples include microscopy, ultrasound, X-ray transmission, positron emission, magnetic resonance,electrical impedance, photoacoustic effect, microwave radiation, atmospheric muons, and seismic
waves. The realm of imaging modalities is constantly expanding. Because of their non-intrusive
nature, each image reconstruction technique requires finding the solution of an ill-posed mathematical
inverse problem to recover the physical properties of a medium using only external measurements.
However, each modality and application considered still provides different challenges from both thetheoretical and computational point of views. This mini-symposium is aiming to highlight some of themost recent, promising and exciting scientific developments to overcome them.

193

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ULTRASOUND DATA AS A PRIOR IN THORACIC IMAGING WITH ELECTRICALIMPEDANCE TOMOGRAPHY

M. Alsaker
Department of Mathematics, Gonzaga University, 502 E. Boone Ave, MSC 2615, Spokane, WA 99258, USA,
alsaker@gonzaga.edu

Abstract
Electrical impedance tomography (EIT) has many potential thoracic imaging applications, but images
suffer from low spatial resolution due to the severe ill-posedness of the inverse problem. On the other hand,ultrasound tomography (UST) has only been minimally studied for use in thoracic imaging, due to difficulties
with wave propagation in lung tissue. However, recent work has indicated low-frequency UST has potential foruse in thoracic imaging, while recent advances in direct D-bar reconstruction methods in EIT have led tostabilized reconstructions with improved resolution. In this talk, we use low-frequency UST data to create aspatial prior for direct D-bar EIT reconstruction, thus combining the two modalities.
This is a joint work with Diego Armando Cardona Cárdenas, Sérgio Shiguemi Furuie, and Jennifer L.
Mueller

194

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARIZATION OF INVERSE PROBLEMS BY FILTERED DIAGONAL FRAME
DECOMPOSITION
Andrea Ebner
Department of Mathematics, University of Innsbruck,Technikerstrasse 13, 6020 Innsbruck, Austria,
andrea.ebner@uibk.ac.at

Abstract
The characteristic feature of inverse problems is their instability with respect to data perturbations. In
order to stabilize the inversion process, regularization methods have to be developed and applied. We introduce
and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singularvalue decomposition to the frame case. Frames as generalized singular system allows to better adapt to a givenclass of potential solutions. We show that filtered diagonal frame decomposition yield a convergent
regularization method. Further, we derive convergence rates under source type conditions and prove order
optimality under the assumption that the considered frame is a Riesz-basis. In addition, we apply our results to a
tomography problem based on the Radon tranform.

References
1. A. Ebner, J. Frikel, D. Lorenz, J. Schwab, and M. Haltmeier. Regularization of inverse problems by filtereddiagonal frame decomposition. ArXiv preprint, August 2020

195

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SIMULTANEOUS RECONSTRUCTION OF EMISSON AND ATTENUATION IN
PASSIVE GAMMA EMISSON TMOGRAPHY OF SPENT NUCLEAR FUEL

Tatiana Bubba
Department of Mathematical Sciences, University of Bath, UKtab73@bath.ac.uk

Abstract
The International Atomic Energy Agency (IAEA) has recently approved passive gamma emission
tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs), an
important aspect of international nuclear safeguards. The PGET instrument resembles a single photon emission
computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission
map of the SFA. The fuel material strongly self-attenuates its gamma-ray emissions, so that correctly accounting
for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it isdesirable to use as little a priori information as possible about the fuel, including the attenuation map, in thereconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform
attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered backprojection (FBP) reconstruction.
We propose a method to simultaneously reconstruct the emission and attenuation maps by formulating
the reconstruction as a constrained minimization problem with a least squares data fidelity term andregularization terms. The performance of the proposed method, with two different regularizers, is evaluated withsimulated and real data that include missing rods and rods replaced with fresh fuel. The
method is shown to produce good results when comparing the reconstructions to the ground truth by various
numerical metrics, and when classifying the rods with the method currently employed by the IAEA.
Additionally, the proposed method is shown to allow for an enhanced classification method that uses also thereconstructed attenuation map.
This is a joint work with R. Backholm, C. Belanger-Champagne, P. Dendooven, T. Helin and S.
Siltanen.

196

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DEEP IMAGE PRIOR RECONSTRUCTION FOR 3D MAGNETIC PARTICLEIMAGING
Tobias Klutha , Sören Dittmera and Peter Maasa
a Center for Industrial Mathematics, University of Bremen, Bibliothekstr. 5, 28359 Bremen, Germany,tkluth@math.uni-bremen.de

Abstract
Magnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect theconcentration of superparamagnetic iron oxide nanoparticles. The forward operator describing the relationshipbetween the tracer concentration and the induced voltage in receive coil units is typically given by a Fredholm
integral operator of the first kind [1]. A spatial encoding in the time-dependent voltage measurement is obtainedby the combination of the nonlinear magnetization behavior of the nanoparticles and the characteristic of theapplied magnetic field comprising a static gradient field and a highly dynamic oscillating field. The image
reconstruction problem in MPI requires the solution of a severely ill-posed problem [2].
Deep image priors (DIP) have been recently introduced as a machine learning approach for tasks inimage processing [3] and its applications to inverse problems [4]. The number of deep learning methods which areapplicable to solve inverse problems is continuously increasing. Commonly these methods follow a to step
procedure. First a neural network is trained by minimizing a certain loss function for a large number of data points
and afterwards new data is fed into the network to obtain a solution to the desired task. In contrast, the DIP
approach is different. A solution is obtained by unsupervised training with one single data point. Given one noisy
data point y  and an ill-posed operator equation Ax  y  , the task of DIP is to train a network (z ) with network parameters  and input z by minimizingA ( z )  y  ,1where the random input z is kept fixed. After training the solution to the inverse problem is obtained byevaluating the network, i.e., x   (z ) . Note that the standard DIP uses an l 2 -norm in the loss function. For
MPI an l1-norm turned out to be beneficial for the reconstruction quality.
In this talk we discuss regularization properties of the DIP method in a specific theoretical setup.
Furthermore, we apply a 3D DIP to MPI and show 3D reconstructions from real MPI data. The DIP approach iscompared to various regularization methods including the minimization of Tikhonov-type functionals with eitherl 2 -discrepancy or l1 --discrepancy terms. Penalty terms include an l 2 -term, an l1 -term, and a TV-term. All
methods are also compared to the standard reconstruction technique in MPI, i.e., minimizing a standard Tikhonovfunctional with a Kaczmarz-type algorithm which in addition includes an implicit regularization mechanism due
to early stopping. The latter mechanism turned out to be beneficial for dealing with the noise characteristic in
MPI. The behavior of all considered methods is illustrated by quantitative (in terms of image quality; PSNR andSSIM) and qualitative results on a publicly available 3D dataset (Open MPI dataset).
This talk is based on joint work with S. Dittmer, D. Otero Baguer, and P. Maass..

References
1. T. Kluth. Mathematical models for magnetic particle imaging. Inverse Problems, 34(8):083001, 2018.
2. T. Kluth, B. Jin, and G. Li. On the degree of ill-posedness of multi-dimensional magnetic particle imaging.
Inverse Problems, 34(9):095006, 2018.
3. V. Lempitsky, A. Vedaldi, and D. Ulyanov. Deep image prior. 2018 IEEE/CVF Conference on Computer
Vision and Pattern Recognition, pp. 9446–9454, 2018.
4. S. Dittmer, T. Kluth, P. Maass, and D. Otero Baguer. Regularization by architecture: A deep prior approach
for inverse problems. Journal of Mathematical Imaging and Vision, 2019.

197

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

GEOPHYSICAL AND MEDICAL IMAGING: WHAT THEY CAN LEARN FROMEACH OTHER

Volker Michel
Geomathematics Group, Department of Mathematics, University of Siegen, Germany
michel – at – mathematik.uni-siegen.de

Abstract
Tomographic imaging problems in geophysics and medicine have a lot of common aspects. The at-firstsight similarities are obvious: we have ill-posed inverse problems with an unknown function on an
approximately ball-shaped domain. However, there are also remarkable connections in the details. We will
investigate this for two inverse problems: the inversion of electro-magnetoencephalography(E-MEG)
data in medicine and the inversion of gravitational observations in geophysics. For example, the representationof these inverse problems as Fredholm integral equations of the first kind contain integral kernels which have
expansions in terms of Legendre polynomials with very similar coefficients. As a consequence, the null spaces
have many common characteristics. Moreover, the well-known downward continuation problem of satellite dataalso contributes to the exponential ill-posedness of E-MEG inversion.
The speaker and his coauthors have shown that these connections can be utilized to extend the theory ofthese inverse problems and to develop advanced and particularly adapted numerical methods. We were able toobtain e.g. previously unknown singular value decompositions and insights on the ill-posedness of the inverse
problems. Moreover, our numerical approaches managed to outperform some previously known methods. Thiswas verified with synthetic, but realistic data examples. Moreover, we also demonstrated the applicability to real
data.
This talk is an attempt to give an overview on some of the results which were obtained on inverse
gravimetry (and the related downward continuation) and on E-MEG inversion.
Parts of this presentation are based on joint works with A.S. Fokas, O. Hauk, S. Leweke, and R.
Telschow. For further details, see [1-10].

References
1. A.S. Fokas, O. Hauk, and V. Michel: Electro-magneto-encephalography for the three-shell model: numerical
implementation via splines for distributed current in spherical geometry, Inverse Problems, 28, 035009 (2012).
2. S. Leweke: The Inverse Magneto-electroencephalography Problem for the Spherical Multiple-shell Model,
PhD Thesis, Department of Mathematics, University of Siegen, 2018.
3. S. Leweke, O. Hauk, and V. Michel: article in preparation.
4. S. Leweke and V. Michel: article in preparation.
5. S. Leweke, V. Michel, A.S. Fokas: Electro-magnetoencephalography for the spherical multiple-shell model:
novel integral operators with singular-value decompositions, Inverse Problems, 36(3) (2019),
https://doi.org/10.1088/1361-6420/ab291f.
6. S. Leweke, V. Michel, R. Telschow: On the non-uniqueness of gravitational and magnetic field data inversion(survey article), in: Handbook of Mathematical Geodesy (W. Freeden and M.Z. Nashed, eds.), Birkhäuser,
Basel, 2018, pp. 883-919.
7. V. Michel: Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution
from data of the Earth's gravitational field at satellite height, Inverse Problems, 21, 997-1025 (2005).
8. V. Michel, A.S. Fokas: A unified approach to various techniques for the non-uniqueness of the inverse
gravimetric problem and wavelet-based methods, Inverse Problems, 24, 045019 (2008).
9. V. Michel, S. Orzlowski: On the null space of a class of Fredholm integral equations of the first kind, J.
Inverse Ill-Posed Poblems, 24, 687-710 (2016).

198

MINISYMPOSIUM
M22: Inverse Problems with Data-Driven Methods and Deep Learning
Organizers:
Tatiana Bubba, University of Bath, UK, tab73@bath.ac.uk
Martin Genzel, Helmholtz-Zentrum Berlin für Materialien und Energie,
Germany, martingenzel@googlemail.com
Andreas Hauptmann, University of Oulu, Finland, andreas.hauptmann@oulu.fi
Maximilian März, Technical University of Berlin, Germany, maerz@math.tu-berlin.de
We are currently experiencing a paradigm shift in image reconstruction. Robust mathematical
inversion algorithms are combined with emerging methods in data science to achieve state-of-the-art
results in a wide range of inverse problems. A successful application of these methods in practice
involves a thorough understanding of their mathematical properties and the underlying imaging
physics. This minisymposium aims to bring experts in data-driven methods and deep learning forinverse problems together and provides an overview of learned image reconstruction approaches,mathematical insights, and real-world applications.

199

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STILL NO FREE LUNCH – ON AI GENERATED HALLUCINATIONS AND THE
ACCURACY-STABILITY TRADE-OFF IN INVERSE PROBLEMS

Vegard Antun
Department of Mathematics, University of Oslo, Norway
vegarant@math.uio.no.

Abstract
AI generated hallucinations are becoming a serious concern in modern methods for image
reconstruction in inverse problem. The phenomenon seems universal and has become apparent in a variety ofmodalities:“The most serious issue when applying deep learning for discovery is that of hallucination. [...] These
hallucinations are deceptive artifacts that appear highly plausible in the absence of contradictory information andcan be challenging, if not impossible, to detect.”– In “Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction”,
Nature Methods (2021) [1].
“The potential lack of generalization of deep learning-based reconstruction methods as well as theirinnate unstable nature may cause false structures to appear in the reconstructed image that are absent in theobject being imaged ”– In “On hallucinations in tomographic image reconstruction”, IEEE Trans. Med.
Imaging (2021) [2].
In this talk, we present a comprehensive mathematical analysis explaining the many facets of AI
generated hallucinations, their links to instabilities, but also how stable AI methods can hallucinate. Our results
establish four crucial issues for AI methods in inverse problems that can be interpreted as ’no free-lunch’phenomena: (1) Overly accurate AI methods will wrongly transfer details from one image to anotherreconstructed image creating a hallucination. Our mathematical theory describes exactly how these details very
naturally can appear in data-driven setups and why this phenomenon occurs. (2) There is an accuracyhallucination barrier: Too accurate reconstructions – on for example the training set – will yield hallucinations,and reducing the accuracy on certain inputs will reduce the hallucinations. (3) There is an accuracy-stability
trade-off, and optimising this trade-off is challenging through standard training processes. (4) Hallucinations canoccur due to any kind of noise model and probability distribution on the data used, and standard training has few
ways of protecting against the conditions leading to AI generated hallucinations.
Finally, our results show that accurate and hallucination-free methods can only be achieved by having
information about the kernel of the sampling operator encoded in the recovery algorithm. Based on this, we
initiate a program for reducing hallucinations in DL in inverse problems. This suggests a path forward toward
hallucination free AI methods in inverse problems.

References[1] David P Hoffman, Isaac Slavitt, and Casey A Fitzpatrick. The promise and peril of deep learning in
microscopy. Nature Metods, 18(2):131–132, 2021.
[2] Sayantan Bhadra, Varun A Kelkar, Frank J Brooks, and Mark A Anastasio. On hallucinations in tomographic
image reconstruction. IEEE Trans. Med. Imaging, 40(11):3249–3260, 2021

200

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A BAYESIAN DEEP IMAGE PRIOR
Riccardo Barbano
Department of Compuer Science, University College London, Gower St, London WC1E 6EA
riccardo.barbano.19@ucl.ac.uk

Abstract
Existing deep learning based tomographic image reconstruction methods do not provide accurate
estimates of reconstruction uncertainty, hindering their real-world deployment. To address this limitation, weconstruct a Bayesian prior for tomographic reconstruction, which combines the classical total variation (TV)regulariser with the modern deep image prior (DIP). Specifically, we use a change of variables to connect ourprior beliefs on the image TV semi-norm with the hyper-parameters of the DIP network. For the inference, wedevelop an approach based on the linearised Laplace method, which is scalable to high-dimensional settings. The
resulting framework provides pixel-wise uncertainty estimates and a marginal likelihood objective for
hyperparameter optimisation. We demonstrate the method on synthetic and real-measured high-resolution μCT
data, and show that it provides superior calibration of uncertainty estimates relative to previous probabilistic
formulations of the DIP.
This is joint work with J Antorán, J Leuschner, JM Hernández-Lobato, B Jin

201

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

GROUND TRUTH FREE DENOISING BY OPTIMAL TRANSPORT
Sören Dittmer
Univeristy of Cambridge, UK & University of Bremen, Germany
sd870@cam.ac.uk

Abstract
This study proposes a new training strategy for a denoiser removing (additive) independent noise, with
only as readily available data as possible and no further assumptions on the data nor noise. While every realworld measurement contains some noise, it seems that this problem remains unsolved for settings where clean
data samples are lacking.
We propose a pushforward operator formulation of an ideal denoiser and a corresponding GAN setup
for training a denoiser ground truth free. The GAN trains solely on samples of noisy data and noise. In a series ofdenoising experiments in 1D and 2D, we demonstrate our training strategy's performance, which significantly
improves the state-of-the-art of unsupervised denoising. Moreover, for some non-Gaussian noise, the method
compares favorably even to naive supervised denoising.

202

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARISING INVERSE IMAGING PROBLEMS USING GENERATIVE DEEPLEARNING METHODS

M. Duffa , N. D. F. Campbellb and M. J. Ehrhardta
a Department of Mathematics,University of Bath, Bath, BA2 7AY, UK, m.a.g.duff@bath.ac.uk
bDepartment ofComputer Science, University of Bath, Bath, BA2 7AY, UK

Abstract
Deep neural network approaches to inverse imaging problems have produced impressive results in thelast few years. We consider the use of generative models in a variational regularisation approach to inverse
problems. Generative models learn, from observations, approximations to high-dimensional data distributions.
The considered regularisers penalise images that are far from the range of a generative model that has learned to
produce images similar to a training dataset. We name this family generative regularisers.
For example, let 𝐴:𝑋→ 𝑌 be a forward process that takes an image 𝑥∈ 𝑋 to data 𝑦∈ 𝑌. The inverse
problem takes data, 𝑦 corrupted by noise, 𝜖, and finds image, 𝑥, such that 𝑦=𝐴(𝑥). A trained generator 𝐺:𝑍→ 𝑋,
takes values in a known lower dimensional latent space, 𝑍 and outputs images similar to some training set. One
common approach to incorporate the generator as part of a regulariser, introduced by Bora et al. [1], searches
through the range of the generator to solve the inverse problem:z *  arg min A(G ( z ))  y , x*  G( z * )

(1)

zZ

Similar ideas have been further explored by [2,3,4].
In contrast to other data driven approaches, generative regularisers do not require paired training data
and are learnt independently of the forward model. This makes the method very flexible in real-world scenarios
where noise levels and forward model parameters may change.
In this talk, we will give numerical examples, using variational autoencoders (VAEs) [5] and generative
adversarial networks (GANs) [6] demonstrating three different approaches of including generative models insolving inverse problems on simple datasets. The success of these methods depends on the quality of a generator.
For example, in a deterministic setting, such as (1), we also require that the ground truth image lies in the rangeof the generator. We will discuss desired criteria for evaluating a trained generative network, in the context of
inverse problems, allowing comparison between generative model approaches and providing direction for future
work.

References
1. A. Bora, A. Jalal, E. Price, and A. G. Dimakis, Compressed Sensing Using Generative Models," inInternational Conference on Machine Learning, 2017.
2. V. Shah and C. Hegde, Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable
Guarantees," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2018.
3. M. Dhar, A. Grover, and S. Ermon, Modeling Sparse Deviations for Compressed Sensing Using Generative
Models," in International Conference on Machine Learning, 2018.
4. D. Obmann, J. Schwab, and M. Haltmeier, Deep synthesis regularization of inverse problems," arXiv preprint
arXiv:2002.00155, 2020.
5. D. P. Kingma and M. Welling, Auto-Encoding Variational Bayes," in International Conference on Learning
Representations, 2014.
6. I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio,
Generative adversarial nets", arXiv:1406:2661, 2014.

203

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

EFFICIENT TRAINING OF INFINITE-DEPTHE NEURAL NETWORKS VIAJACOBIAN-FREE BACKPROPAGATION

S. Wu Funga , H. Heatonb , Qiuwei Lic, Daniel McKenzied, Stanley Osherd, and Wotao Yinca
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado, USA.
swufung@mines.edubTypal Research, Los Angeles, California, USA, research@typal.llc
cAlibaba Group (USA) Damo Academy, Los Angeles, California, USAd
Department of Mathematics, University of California, Los Angeles, USA

Abstract
A promising trend in deep learning replaces fixed depth models by approximations of the limit as
network depth approaches infinity. This approach uses a portion of network weights to prescribe behavior bydefining a limit condition. This makes network depth implicit, varying based on the provided data and an error
tolerance. Moreover, existing implicit models can be implemented and trained with fixed memory costs in
exchange for additional computational costs. In particular, backpropagation through implicit networks requires
solving a Jacobian-based equation arising from the implicit function theorem. We propose a new Jacobian-freebackpropagation (JFB) scheme that circumvents the need to solve Jacobian-based equations while maintaining
fixed memory costs. This makes implicit depth models much cheaper to train and easy to implement. Numericalexperiments on classification and medical imaging are provided.

References
1. S. Wu Fung, H. Heaton, Q. Li, D. McKenzie, S. Osher, and W. Yin, JFB: JFB: Jacobian-Free
Backpropagation for Implicit Networks. Proceedings of the AAAI Conference on Artificial Intelligence,(2022).

204

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MEASURING AND ENHANCING ROBUSTNESS IN DEEP LEARNING BASEDCOMPRESSIVE SENSING
Mohammad Zalbagi Darestani, Jiayu Liu, Akshay Chaudhari and Reinhard Heckel
Rice University, Stanford University, Technical University of Munich, email: Reinhard.heckel@tum.de

Abstract
Deep-learning based algorithms outperform traditional, handcrafted algorithms for reconstructing
images from few and noisy measurements. However, neural networks may be sensitive to small, yetadversarially-selected perturbations, may perform poorly under distribution shifts, and may fail to recover small
but important features in an image.
To understand the sensitivity to such perturbations, we measured the robustness of a variety of deep
network based and traditional methods. Perhaps surprisingly, in the context of accelerated magnetic resonanceimaging, we find no evidence that deep learning based algorithms are less robust than classical, un-trained
methods. Even for natural distribution shifts, we find that classical algorithms with a single hyper-parameter
tuned on a training set compromise as much in performance than a neural network with 50 million parameters.
Our results indicate that the state-of-the-art deep-learning-based image reconstruction methods provide improved
performance than traditional methods without compromising robustness.
Finally, we present a test-time-training method to improve the distributional robustness of deep-learning
based imaging.

References
1. Mohammad Zalbagi Darestani, Akshay Chaudhari, and Reinhard Heckel, "Measuring Robustness in Deep
Learning Based Compressive Sensing." International Conference of Machine Learning, 2021.
2. Mohammad Zalbagi Darestani, Jiayu Liu, and Reinhard Heckel, ``Test-Time Training Can Close the Natural
Distribution Shift Performance Gap in Deep Learning Based Compressed Sensing’’, arXiv:2204.07204, 2022.

205

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STOCHASTIC NORMALIZING FLOWS FOR INVERSE PROBLEMS:
A MARKOV CHAINS VIEWPOINT

Johannes Hertrich
TU Berlin, Germanyj.hertrich@math.tu-berlin.de

Abstract
Normalizing flows are invertible neural networks which aim to learn a diffeomorphism pushing forward
the normal distribution onto a more complicated distribution. They can be used for generative modeling and for
estimating probability density functions from given samples. However, recently it was shown that learning
multimodal or heavy tailed distributions with normalizing flows requires exploding Lipschitz constants. Thus,
using normalizing flows for generative modeling requires large architectures and the training suffers frominstabilities.
To overcome these topological constraints and to improve the expressiveness of normalizing flowarchitectures, Wu, Köhler and Noé introduced stochastic normalizing flows which combine deterministic,
learnable flow transformations with stochastic sampling methods. In this talk, we consider stochastic
normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general
Markov kernels and which allows to incorporate distributions without densities in a sound way. Here, we derivea loss function based on the Kullback-Leibler divergence and Radon Nikodym derivatives. Further, we show thatstochastic normalizing flows build a unifying framework for a wide range of powerful generative models as
variational autoencoders and diffusion normalizing flows. Thus, our framework establishes a usefulmathematical tool to combine the various approaches. For tackeling Bayesian inverse problems, we extend theideas of conditional and stohastic normalizing flows for sampling from the posterior distribution.
Finally, we demonstrate the performance of stochastic normalizing flows for some numerical examples.
This is a joint work with Paul Hagemann and Gabriele Steidl.

206

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOLVING MMV PROBLEMS VIA ALGORITHM UNFOLDINGPeter Jung
Electrical Engineering and Computer Science, Communications and Information Theory Chair, Technical
University Berlin, Einsteinufer 25, 10587 Berlin, Germany
peter.jung@tu-berlin.de

Abstract
Recovering structured data from compressive observations is a core task many fields, likecomputational imaging, communications and information processing. These problems are ill-posed and priorassumptions are necessary to restrict the solutions.
A particular instance is the multiple measurement vector (MMV) problem where one wants to sense
joint-sparse vectors from a few compressive and repeated observations, far less than its ambient dimension.
Fundamental works show that under certain conditions this can be provably achieved in a robust and stablemanner with computationally tractable algorithms.
For real-world problems, however, it is difficult and often impossible to analytically treat detailed
structure beyond joint-sparsity and to perform optimal algorithm tuning. Therefore, recovery approaches, wellunderstood in theory, perform often suboptimal in practice. Algorithms converge slowly and increased
acquisition time and sampling rates are necessary to achieve a given target resolution. Approaching highdimensional applications under almost real-time conditions is therefore often not feasible.
In this talk I will present joint works with J. Hauffen and O. Musa on recent unfolding approaches for
the MMV problem. Only few iterations of an algorithm are unrolled into a neural network and the parameters aretrained data-driven via gradient descent and back-propagation. However, learning all weights in a highdimensional setting requires a large amount of training data and in many application training data is scarce.
Therefore, I will discuss here the case of analytic weight computation which tremendously reduce
training overhead and makes this approach more accessible to many new applications.
Furthermore, as in the single vector case, this even permits a theoretical treatment of the trained architecture. The
results will be more refined to convolutional models and then discussed in the context applications in photothermal imaging and in wireless communication.

207

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECOVERY ANALYSIS FOR PLUG-AND-PLAY PRIORS USING THE
RESTRICTED EIGENVALUE CONDITION
U.S. Kamilov
Washington University in St. Louis, USA
kamilov@wustl.edu

Abstract
The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used
for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging
performance and the theoretical convergence properties of these algorithms have been widely investigated, theirrecovery properties have not previously been theoretically analyzed. We address this gap by showing how toestablish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the
fixed-points of a deep neural network. We also present numerical results comparing the recovery performance of
PnP/RED in compressive sensing against that of recent compressive sensing algorithms based on generative models.
Our numerical results suggest that PnP with a pre-trained artifact removal network provides significantly better
results compared to the existing state-of-the-art methods.

208

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

2DETECT - A LARGE 2D EXPANDABLE, TRAINABLE, EXPERIMENTALCOMPUTED TOMOGRAPHY DATA COLLECTION FOR MACHINE LEARNING

Maximilian Kiss
CWI, Netherlands
Maximilian.Kiss@cwi.nl

Abstract
Recent research in the computational imaging community largely focuses on the use of machinelearning (ML) techniques for tomographic image reconstruction. Realistic experimental data, i.e. suitableprojection datasets with high-quality ground truth reconstructions and/or segmentations are scarce, but highly
important for the development and training of deep learning algorithms. Our open data collection fills this gap
and provides the community with a versatile 2D CT dataset for machine learning. For this, a diverse mix ofsamples with high natural variability in both, inter and intra-sample shape and density, were scanned with alaboratory X-ray set-up. This sample mix of dried fruits and nuts was filled in a carboard tube (height 34cm,
diameter 10cm) and scanned slice by slice in a fan-beam acquisition setting. More than 5000 slices consisting of3600 projections per full orbit were scanned in two beam settings, reconstructed with highest quality, andsupplemented with a ground truth segmentation. We provide the complete image reconstruction pipeline: raw
projection data (sinograms), a description of the scanning geometry, pre-processing and reconstruction scripts
using open software, and the reconstructed slices as well as their ground truth segmentations. Due to this, thedataset can be used not only for ML-based reconstruction or segmentation in various settings such as limited orsparse angle, but also for noise reduction for general algorithm development.

209

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOLVING INVERSE PROBLEMS WITH DEEP NEURAL NETWORKS ROBUSTNESS INCLUDEDJan Macdonald
Berlin Technical University of Berlin, Germany
macdonald@math.tu-berlin.de

Abstract
In the past five years, deep learning methods have become state-of-the-art in solving various inverse
problems. Before such approaches can _nd application in safety-critical fields, a verification of their reliability
appears mandatory. Recent works have pointed out instabilities of deep neural networks for several image
reconstruction tasks. In analogy to adversarial attacks in classification, it was shown that slight distortions in the
input domain may cause severe artifacts. We shed new light on this concern, by presenting an extensive study of
the robustness of deep-learning-based algorithms for solving underdetermined inverse problems. This covers
compressed sensing with Gaussian measurements as well as image recovery from Fourier and Radon
measurements, including a real-world scenario for magnetic resonance imaging (using the NYU-fastMRI
dataset). Our main focus is on computing adversarial perturbations of the measurements that maximize thereconstruction error. A distinctive feature of our approach is the quantitative and qualitative comparison withtotal-variation minimization, which serves as a provably robust reference method. In contrast to previous
findings, our results reveal that standard end-to-end network architectures are not only resilient against statisticalnoise, but also against adversarial perturbations.
All considered networks are trained by common deep learning techniques, without sophisticated defense
strategies.
This is joint work with Martin Genzel and Maximilian Marz.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DATA-DRIVEN ADVERSARIAL REGULARIZATION FOR IMAGING INVERSE
PROBLEMS

Subhadip Mukherjee
University of Cambridge, UK
sm2467@cam.ac.uk

Abstract
Variational reconstruction has by far been the most widely used approach for solving ill-posed inverse
problems in imaging applications. Nevertheless, the impressive success of deep learning in recent layers has led
to the development of a number of data-driven methods that significantly outperform the classical variational
approaches in terms of reconstruction quality. Such data-driven approaches, although numerically superior to theclassical variational approaches with hand-crafted regularizers, generally come with no theoretical stability andconvergence guarantees. Within the realm of learning-based approaches, there exists a particularly notable lineof research that seeks to model the regularizer in the variational setting using a deep neural network and learn theparameters using an ensemble of training data.
In this talk, we will introduce some new approaches based on adversarial learning to construct datadriven variational regularizers that are amenable to stability analysis and lead to scalable and provably
convergent reconstruction algorithms. More specifically, we will (i) touch upon the problem of learning a convex
regularizer using input-convex neural networks and (ii) show how to combine iterative unrolling with datadriven regularization to achieve fast, high-quality, and provably stable reconstruction

211

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

LEARNED SPEED OF SOUND CORRECTION FOR PHOTOACOUSTICTOMOGRAPHY
Jenni Poimalaa and Andreas Hauptmanna,baResearch Unit of Mathematical Sciences, University of Oulu,Oulu, Finland, jenni.poimala@oulu.fi;bDepartment of Computer Science, University College London, London, UK, andreas.hauptmann@oulu.fi;

Abstract
Photoacoustic tomography is a hybrid imaging modality that combines optical contrast with ultrasonicdetection, and it has shown the potential in a variety of biomedical applications. Real-time applications in threedimensional photoacoustic tomography are relying on fast reconstruction algorithms, instead of utilising fullwave solvers. Unfortunately, the majority of available fast solvers assume homogenous speed of sound (SoS)
distributions. Additionally, reconstruction quality depends on the correct choice for a constant SoS in the imaged
target. In practical experiments, accurate knowledge of the SoS of the imaged target is commonly not available.
To overcome these limitations, we propose a data-driven correction to compensate for unknown orheterogeneous SoS distributions. Especially, we evaluate the potential to include a learned model correction in kspace into the fast FFT based reconstruction algorithm in combination with a learned post-processing andcompare to post-processing alone.
The feasibility of the approaches is studied with simulated data and human in vivo measurements.

212

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SOLVING ILL-POSED INVERSE PROBLEM WITH PRETRANED DENOISERS,GANS AND SUPER-RESOLVERS: THE BP TERM AND THE CORRECTIONFILTERT. Tirer, S. Abu-Hussein, R. Giryes
School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel, tomtirer@mail.tau.ac.il

Abstract
Ill-posed linear inverse problems appear in many image processing applications, such as deblurring,super-resolution and compressed sensing. Many restoration strategies involve minimizing a cost function, whichis composed of fidelity and prior terms, balanced by a regularization parameter. While a vast amount of researchhas been focused on different prior models, the fidelity term is almost always chosen to be the least squares (LS)objective, that encourages fitting the linearly transformed optimization variable, Ax , to the observations y : LS ( x, y ) 12
y  Ax 2 .
2(1)

In this talk, we will propose and examine a different fidelity term, dubbed the back-projection (BP)term, which “back-projects” the residual to the signal space using the (possibly regularized) pseudo-inverse( A ) of the linear operator:

 BP ( x, y ) 2
1A ( y  Ax) .
2
2

(2)

Using the simple proximal gradient method with the BP term and off-the-shelf CNN denoisers or GANs
(that impose the prior) gives excellent empirical results and requires less parameter tuning than LS-based
methods [1]. Moreover, I will present several theoretical results showing cases with advantages of BP over LSboth in accuracy and in convergence rate [2], [3].
While using off-the-shelf CNN denoisers and GANs in inverse problems has recently become
extremely popular, it is much less common to use other more “task-specific” networks for observation models
that have not been considered in their training phase. For example, state-of-the-art CNN super-resolvers sufferfrom a huge performance drop when the downscaling kernel in test time even slightly mismatches the kernel thatis used for training the networks.
We will show how to mitigate this issue for off-the-shelf CNN super-resolvers, which are (typically)
trained with the bicubic downscaling kernel (without any change in the networks). Specifically, inspired by theliterature on generalized sampling, we design a correction filter h that modifies the low-resolution image tomimic one which is obtained by another kernel (e.g., the bicubic) [4]. Thus, applying existing pre-trained superresolvers on the “corrected” low-resolution image, h * y , significantly improves their results. We also show that
our approach outperforms other super-resolution methods that are designed for general downscaling kernels. hhyReferences
1. T. Tirer and R. Giryes, "Image restoration by iterative denoising and backward projections." IEEE
Transactions on Image Processing 28.3 (2018): 1220-1234.
2. T. Tirer and R. Giryes, "Back-projection based fidelity term for ill-posed linear inverse problems." IEEE
Transactions on Image Processing 29 (2020): 6164-6179.
3. T. Tirer and Raja Giryes. "On the convergence rate of projected gradient descent for a back-projection based
objective." SIAM Journal on Imaging Sciences (2021).
4. S. Abu-Hussein, T. Tirer, and R. Giryes. "Correction filter for single image super-resolution: Robustifying offthe-shelf deep super-resolvers." CVPR 2020.

213

MINISYMPOSIUM
M23: Theory and Numerics for Inversion Strategies
Organizer:
Gen Nakamura, Hokkaido University, Japan, nakamuragenn@gmail.com
In recent decades, the inverse problem community has witnessed remarkable developments inboth theoretical and numerical aspects. In this mini-symposium, we are aiming to bring together many
experts mainly from Asia who are working in a wide range of inverse problems and their related fields
concerning some inversion strategies for inverse problems. This will give us a good opportunity toshare some motivations, backgrounds and viewpoints of several inverse problems. Further, it will also
improve the communication and collaboration especially between researchers in Asia and Europe.

214

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

LOCAL RECOVERY OF A PIECEWISE CONSTANT ANISOTROPICCONDUCTIVITY IN EIT ON DOMAINS WITH EXPOSED CORNERS

Takashi Furuya
Department of Mathematics, Hokkaido University, Japan
takashi.furuya0101@gmail.com

Abstract
We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric
impedance tomography) on certain bounded Lipschitz domains $\Omega$ in $\mathbb{R}^2$ with corners. The
measurement is conducted on a connected open subset of the boundary $\partial\Omega$ of $\Omega$containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity isdefined via a decomposition of $\Omega$ into polygonal cells. Specifically, we consider a parallelogram-based
decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the
conductivity on each cell is unknown. We prove the local recovery near a known piecewise constant anisotropic
conductivity $\gamma_0$. We do so by proving the injectivity of the Fr\'echet derivative $F'(\gamma_0)$ of theforward map $F$, say, at $\gamma_0$. The proof presented, here, involves defining different classes ofdecompositions for $\gamma_0$ and a perturbation or contrast $H$ in a proper way so that we can find in theinterior of a cell for $\gamma_0$ exposed single or double corners of a cell of $\mbox{\rm supp}H$ for theformer decomposition and latter decomposition, respectively. Then, by adapting the usual proof near suchcorners, we establish the mentioned injectivity.
This is a joint work with Maarten de Hoop (Rice University), Ching-Lung Lin (National Cheng-Kung
University), Gen Nakamura (Hokkaido University), Manmohan Vashith (Indian Institute of Technology).

References
1. M. V. de Hoop, T. Furuya, C. L. Lin, G. Nakamura, M. Vashisth, ``Local recovery of a piecewise constantanisotropic conductivity in EIT on domains with exposed corners'', arXiv:2202.06739, (2022)

215

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

EXTRACTION OF THE MASS DENSITY BY EMBEDDING CONTRASTED SMALL
INCLUSIONS
Durga Prasad Challa a, Divya Gangadaraiaha, Mourad SinibaIndian Institute of Technology, Tirupati, India, durga.challa@iittp.ac.in ; ma18d501@iittp.ac.in ;bAustrian Academy of Sciences, Austria, mourad.sini@oeaw.ac.in;

Abstract
We deal with the time-harmonic elastic wave propagation in the presence of the small-scaled particles of highrelative mass density. These particles generate few local spots at their locations and are generated as the possible
body waves related to elastic resonances. We can observe that the eigenvalues of the elastic Newtonian potential
and a family of the mentioned resonances are related.
Our aim is twofold. First, to characterize the dominant scattered fields due to the injection of small particles
for incident frequencies near resonances. Second, to tackle the inverse problem of reconstructing the massdensity.
The main idea in achieving our goal is to invert the associated Lippmann Schwinger system by deriving
required apriori estimates of the total fields to get the dominant field, which further allows us to extract the massdensity. This procedure has advantages in comparison with the classical problem based on the D-N map.

216

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

CARLEMAN ESTIMATES AND INVERSE PROBLEMS FOR THE COUPLEDQUANTITATIVE THERMO-ACOUSTIC EQUATIONS

Michel Cristofol, Shumin Li and Yunxia Shang
University of Science and Technology of China, China, shuminli@ustc.edu.cn

Abstract
We investigate the so called quantitative thermo-acoustic tomography process (e.g. [1, 3, 10, 11] andtheir references). According to [3], assuming that the variations in temperature and pressure are weak andneglecting the nonlinear effects, we obtain the following system:
 2
1

2
 t p  v s div  p    t div( )   t   , in
(1)
Q




   1 div( )   0  p    , t
C p
C p t
C p


for the temperature rise  and the pressure perturbation p from the equilibrium steady state
depending on ( x, t ) : ( x1 ,..., xn , t )  Q :   (0,T ).
We set ( x, t )   t ( x, t ). Differentiating the second equation in (1) with respect to t , weobtain:2 t p  1 ( x) divq ( x )p    2 ( x) div ( x)   ( x) t   , in Q ,2 t    3 ( x) div ( x)    4 ( x) t p   3 ( x) t   ,(2)where 1 ( x)  ( x)vs2  ( x) , q( x) 1/  ( x) ,  2 ( x)  ( x) ,  3 ( x) 1 /(  ( x))C p ( x)and  4 ( x)  0 ( x) ( x) 3 ( x) .

We will establish Carleman estimates for (2) with the exponential weight function exp( 2s ) where ( x, t )  exp( ( x, t )) ,  ( x, t )  x  x0   t 2 , ( x, t )  Q
  0 is a suitably large constant, and   0 0 is a suitably small constant, provided that K ,  , C p , 0 ,  0 and  satisfy some boundedness and positivity conditions. By these Carleman estimates and the2

Bukhgeim-Klibanov method (e.g., [2, 4, 5, 7, 8, 9] and their references), we will consider several inverse
problems for (2).

References[1] Akhouayri H, Bergounioux M, Silva A D, Elbau P, Litman A and Mindrinos L., J. Inverse Ill-Posed Probl.
25 (2016) 703-17.
[2] Bukhgeim A L and Klibanov M V Soviet Math. Dokl. 24 (1981) 244-247.
[3] Cox B and Beard P C, In Wang L V, Ed., Photoacoustic Imaging and Spectroscopy, ch. 3, pp. 25-34. CRC
Press, 2009.
[4] Imanuvilov O Y, Isakov V and Yamamoto M, Pure Appl. Math. 56 (2003) 1366-1382.
[5] Imanuvilov O Y and Yamamoto M, Inverse Probl. 19 (2003) 157-171.
[6] Klibanov M V and Yamamoto M, Appl. Anal. 85 (2006 )515-538.
[7] Isakov V, Inverse problems for partial di_erential equations. Berlin:Springer-Verlag, 2006.
[8] Klibanov M V, Inverse Probl. 8 (1992) 575-596.
[9] Klibanov M V, J. Inverse Ill-Posed Probl. 21(2013) 477-560.
[10] Patch S K and Scherzer O, Guest eds Inroduction: Photo-and thermo-acoustic imaging Inverse Problems 23(2007) S1-S10.
[11] Stefanov P and Uhlmann G, Inverse Problems 25 (2009) 075011.

217

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DIFFUSE OPTICAL TOMOGRAPHY WITH A SIMULATED ANNEALING MONTECARLO ALGORITHM
Yu Jianga , Manabu Machidab and Morikazu TodorokiaSchool of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China,
jiang.yu@mail.shufe.edu.cn;bInstitute for Medical Photonics Research, Hamamatsu University School of Medicine, Hamamatsu 431-3192,
Japan; machida@hama-med.ac.jp;
cDepartment of Physics, Chiba Institute of Technology, Chiba 275-0023, Japan;
todoroki.norikazu@p.chibakoudai.jp

Abstract
Let us consider the following diffusion equation:
 D0 u   0 u  f , x  , D0  u  u  0, x  ,

We will consider the inverse problem of determining  a (x) from boundary measurements.
We propose a Monte Carlo numerical method for this diffuse optical tomography which uses the singlecomponent Metropolis-Hastings algorithm. After a spin Hamiltonian is introduced in the cost function, theinverse problem is solved by simulated annealing.
One of reconstructed images is shown below.

Two absorbers with separation 20 mm.

218

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

UNIQUENESS FOR THE INVERSE BOUNDARY VALUE PROBLEM OFPIECEWISE HOMOGENEOUS ANISOTROPIC ELASTICITY IN THE TIMEDOMAINGen Nakamura
Hokkaido University, Japan
nakamuragenn@gmail.com

Abstract
We consider the inverse boundary value problem of recovering a piecewise homogeneous elastic tensor
and a piecewise homogeneous mass density from a localized lateral Dirichlet-to-Neumann or Neumann-toDirichlet map, for the elasticity equation in the space-time domain.
We derive uniqueness for identifying this tensor and density on alldomains of homogeneity that may bereached from the part of the boundary where the measurements are taken by a chain of subdomains whose
successive interfaces contain a curved portion.

219

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE SOURCE PROBLEMS FOR WAVE EQUATIONS ON LORENTZIANMANIFOLDS

Hiroshi Takase
Institute of Mathematics for Industry, Kyushu University, Japanhtakase@imi.kyushu-u.ac.jp

Abstract
One obtains a system of wave equations with time-dependent principal parts by linearizing the Einsteinequation. Due to the time dependence, it has not been sufficient to investigate inverse problems for such kinds ofhyperbolic equations in Euclidean spaces. Therefore, we introduce geometric analysis on semi-Riemannian
manifolds, especially Lorentzian manifolds, and prove conditional local Hölder stability for inverse source
problems. I will show Sobolev spaces on manifolds, semigeodesic coordinate systems, and Carleman estimateson manifolds.

References
1. H. Takase. Inverse source problem for a system of wave equations on a Lorentzian manifold. Commun.
Partial Differential Equations, 45(10):1414–1434, 2020.

220

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON THE IDENTIFICATION OF SOURCE TERM IN THE HEAT EQUATION FROMSPARSE DATA

Zhidong Zhang
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China

Abstract
We consider the recovery of a source term f ( x, t )  p( x)q(t ) for the nonhomogeneous heat
equation in   (0, ) where  is a bounded domain in R 2 with smooth boundary  fromoverposed lateral data on a sparse subset of   (0, ) . Specifically, we shall require a small finite
number N of measurement points on  and prove a uniqueness result; namely the recovery of the pair
( p; q) within a given class, by a judicious choice of N  2 points. Naturally, with this paucity ofoverposed data, the problem is severely ill-posed.
Nevertheless we shall show that provided the data

221

MINISYMPOSIUM
M24: Inverse Problems via Topological Derivatives
Organizer:
María-Luisa Rapún, Universidad Politécnica de Madrid, Spain, marialuisa.rapun@upm.es
The topological derivative of a shape functional measures the sensitivity of such functional to having an
infinitesimal perturbation at each point of the explored region. It has many applications in connection
with inverse problems such as shape optimization, topology optimization, imaging processing, crack anddefect detection in non-destructive testing, to mention a few.
The aim of this minisymposium is to bring together experts and young researchers working in this field todiscuss and review the recent applications, new results and future challenges.

222

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

TOPOLOGICAL DERIVATIVE BASED BAYESIAN INFERENCE FOR INVERSESCATTERING PROBLEMS

Ana Carpio
Universidad Complutense de Madrid, Spain
carpio@mat.ucm.es

Abstract
Inverse scattering techniques seek to infer the structure of objects integrated in an ambient medium
from data recorded at a set of receptors, which represent some scattered incident radiation.
Solving the inverse problem amounts to finding objects minimizing the difference between the synthetic data
generated by the approximate objects as predicted by a forward model and the true data. When the magnitude of
the noise in the data is small, algorithms combining iteratively regularized Gauss-Newton schemes with
topological derivative based initial guesses and updates of the number of objects may provide reasonable
reconstructions. However, estimating uncertainties inherent to this process as themagnitude of the noise increases is a challenging task.
We propose a topological derivative based Bayesian inference framework.
Numerical simulations illustrate the resulting predictions in light and acoustic holography set-ups.

223

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECONSTRUCTION OF SHARP INTERFACES IN TIME-DOMAIN FULLWAVEFORM INVERSION

Yuri Flores AlbuquerqueInstituto de Matematica e Estatıstica, Universidade de Sao Paulo, Rua do Matao 1010, 05508-090,
Sao Paulo, Brazil
yuri.falbu@gmail.com

Abstract
Most optimization approaches for reconstruction of velocity models used in FWI are based on smoothtechniques such as the Tikhonov regularization. However, realistic velocity profiles usually present various
discontinuities and sharp interfaces, for instance in the case of salt bodies. Thus, approaches based on nonsmooth
optimizations techniques should be used to obtain sharper reconstructions.
In this work, we propose a novel shape optimization based algorithm to track sharp interfaces of thevelocity model for time-domain FWI. The interface of a salt domain is considered as the variable of a trackingtype cost functional. A damping term in the neighborhood of the boundary is used to model an unboundeddomain. Using a Lagrangian approach, we compute the shape derivative of the cost functional. The shapederivative depends on an adjoint state, which is the solution of a wave equation with the residual on right-handside. The interface evolution is performed using a level set method. Using synthetic measurements, we show theefficiency of the method through several examples of reconstruction.

224

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

TOPOLOGICAL SENSITIVITY ANALYSIS FOR IDENTIFICATION OF VOIDSUNDER NAVIER’S BOUNDARY CONDITIONS IN LINEAR ELASTICITY

Amel Ben Abda* and Bochra Mejri***
Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis, LR99ES20 Modélisation Mathématiqueet Numérique dans les Sciences de l’Ingénieur, 1002, Tunis, Tunisie**
University of Côte d'Azur, CNRS, LJAD, Parc Valrose, 28 Avenue Valrose, 06108 Nice Cedex 02, France

Abstract
This work is concerned with a geometric inverse problem related to the two-dimensional linear
elasticity system. Thereby, voids under Navier’s boundary conditions are reconstructed from the knowledge ofpartially over-determined boundary data. The proposed approach is based on the so-called energy-like error
functional combined with the topological sensitivity method. The topological derivative of the energy-like misfit
functional is computed through the topological-shape sensitivity method. Firstly, the shape derivative of thecorresponding misfit function is presented briefly from previous work Méjri (2018 J. Inverse Ill-Posed
Problems). Then, an explicit solution of the fundamental boundary-value problem in the infinite plane with acircular hole is calculated by the Muskhelishvili formulae. Finally, the asymptotic expansion of the topologicalgradient is derived explicitly with respect to the nucleation of a void.
Numerical tests are performed in order to point out the efficiency of the developed approach.

225

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DETECTION OF SMALL CARDIAC ISCHEMIC REGIONS FROM BOUNDARYMEASUREMENTS VIA TOPOLOGICAL GRADIENT

Luca Ratti
Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genova, Italy;luca.ratti@unige.it

Abstract
The monodomain model is one of the most famous mathematical descriptions of the cardiac
electrophysiology, and it consists of a system of a semilinear parabolic partial differential equation withnonlinear ordinary differential equations. Our aim is to detect the presence of small ischemic regions, namely, a
perturbation in the coefficients of the model, from boundary data.
After an analysis of the well-posedness of the model we determine an asymptotic expansion of theperturbed potential due to the presence of small conductivity inhomogeneities and use it to compute thetopological gradient of a suitable boundary misfit functional.
The stability of the reconstruction, even in the case of noisy measurements, is confirmed by several
numerical experiments.
This is joint work with Elena Beretta (Politecnico di Milano – NYU Abu Dhabi) and Cecilia Cavaterra(Università Statale di Milano).

226

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A REAL-TIME IDENTIFICATION OF SMALL CONDUCTIVITY INHOMOGENEITY
VIA TOPOLOGICAL DERIVATIVE
Won-Kwang Park
Department of Information Security, Cryptology, and Mathematics, Kookmin University, 02707, Seoul, Korea,
parkwk@kookmin.ac.kr

Abstract
There exists a considerable amount of interesti.ng inverse problems concerned with the retrieval of inclusions
embedded in a medium from measured boundary data. This is an old but very interesting to nowadays scientists
due to its wide applicability such as detection of cracks in material engineering, ultrasound imaging in medical
sciences, and scanning anti-personnel mines hidden in the ground in military services. Throughout various
researches, remarkable reconstruction techniques have been investigated, most of which are based on the
Newton-type iteration scheme. For a successful application of iteration based scheme, the one must begin withan initial guess that is close to the true one in order to avoid undesirable situations such as non-convergence or
falling into the local minima. Therefore, development of a mathematical theory and an algorithm for generating agood initial guess is an important research topic.
Topological derivative (TD) concept is a non-iterative imaging strategy. This was originally developed forthe shape optimization problem, but its application to rapid shape identification has only recently been proven.
Related works can be found in [1,2,3,4] and references therein. Recently, TD was applied to the inverse
conductivity problems as an iterative scheme [5,6,7]. Following [8], it has been confirmed that TD can beapplied as a non-iterative scheme but an estimation of total number of anomalies is needed.
In this contribution, we apply TD for identifying single, small conductivity inhomogeneity without any apriori information. For this, we design a TD based indicator function with two-different boundary conditions andanalyze its mathematical structure. To explain the feasibility and limitation, simulation results with noisy data
are exhibited.

References
1. M. Bonnet, B. Guzina, and N. Nemitz, Fast non-iterative methods for defect identification, Rev. Eur. Mecan.
Num., 17, 571-582 (2008).
2. H. Ammari, J. Garnier, V. Jugnon, and H. Kang, Stability and resolution analysis for a topological derivative
based imaging functional, SIAM J. Control. Optim., 50, 48-76 (2012).
3. W.-K. Park, Topological derivative strategy for one-step iteration imaging of arbitrary shaped thin, curve-likeelectromagnetic inclusions, J. Comput. Phys., 231,1426-1439 (2012).
4. F. L. Loüer, M.-L. Rapún, Topological sensitivity for solving inverse multiple scattering problems in 3D
electromagnetism. Part I: one step method, SIAM J. Imag. Sci., 10, 1291-1321 (2017).
5. A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical
impedance tomography, Inverse Problems, 28, Article ID: 095010 (2012).
6. M. Hintermüller, A. Laurain, and A. A. Novotny, Second-order topological expansion for electrical
impedance tomography, Adv. Comput. Math, 36, 235-265 (2012).
7. F. Mendoza and S. Keeling, A two-phase segmentation approach to the impedance tomography problem,
Inverse Problems, 33, Article ID: 095010 (2017).
8. A. D. Ferreira and A. A. Novotny, A new non-iterative reconstruction method for the electrical impedance
tomography problem, Inverse Problems, 33, Article ID: 035005 (2017).

227

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

TOPOLOGICAL DERIVATIVE ALGORITHM TESTED AGAINST EXPERIMENTAL
DATA: THE THREE-DIMENSIONAL FRESNEL DATABASE
A. Carpioa , M. Penab, M.L. Rapúnc
a UCM, F. de Ciencias Matemáticas, Plaza de Ciencias 3, 28040 Madrid, Spain, carpio@mat.ucm.es;
bUPM, ETSIAE, Plaza del Cardenal Cisneros 3, 28040, Madrid, Spain, manuel.pena@upm.es;
c UPM, ETSIAE, Plaza del Cardenal Cisneros 3, 28040, Madrid, Spain, marialuisa.rapun@upm.es

Abstract
The Fresnel databases [1, 2] were conceived as tools for benchmarking different inversion algorithms,
as well as providing experimental data to research groups that lack the facilities or laboratories to generate theirown.
They consist of several files with measurements of the electromagnetic field for different
electromagnetic experiments. In these experiments, several dielectric or metallic objects were irradiated fromdifferent positions and frequencies while the electric field was recorded on a set of antennas.
In [3] an algorithm based on the topological derivative [4] was tested against the Fresnel two- and threedimensional databases with very good results. The algorithm consists in computing the topological derivative ofa functional measuring the difference between the recorded measures and the ones expected when a known
object is present. The topological derivative is a field which gives an idea of how sensitive is that functional tothe effect of having an inclusion or a hole in each point in the domain.
With this in mind, the points where the topological derivative attains the largest negative values are
identified as belonging to the actual objects.
The formulae for the computation of the topological derivative both for the dielectric and the
conductive cases were already obtained in [5], where the same algorithm was tested against numericallygenerated data.
In this talk, the results from [3] will be shown, together with a brief explanation of the algorithm used.
We will take special care in describing the results of the three-dimensional database [2] . The processingof this database was much more demanding as the computational cost was higher and also because thefull three-dimensional shape of the objects had to be recovered given only measures taken on the equatorial
plane.
Nevertheless, the topological derivative algorithm proved to be a very suitable method for solving thiskind of inverse problem, as it is a fast and robust method, with no need of any kind of prior information. In thissense, its performance was comparable to that of the other methods tested against the same database.
Finally we will also comment on several possible improvements to the method, some of them areongoing work.

References
1. K. Belkebir and M. Saillard, Testing inversion algorithms against experimental data, Inverse problems, 17,1565 (2001).
2. J.M. Geffrin and P. Sabouroux, Continuing with the Fresnel database: experimental setup and improvements
in 3D scattering measurements, Inverse Problems, 25, 024001 (2009).
3. A. Carpio, M. Pena and M.L. Rapún, Processing the 2D and 3D Fresnel experimental databases via
topological derivative methods, Inverse Problems, 37, 105012 (2021).
4. J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM journal on
control and optimization, 37, 1251-1272 (1999).
5. F. Le Louër and M.L. Rapún, . Topological sensitivity for solving inverse multiple scattering problems inthree-dimensional electromagnetism. Part I: one step method. SIAM Journal on Imaging Sciences, 10, 12911321 (2017).

228

MINISYMPOSIUM
M25: Inverse Problems in Biomedical and Material Imaging
Organizers:
Damiana Lazzaro, University of Bologna, Italy, damiana.lazzaro@unibo.it
Andrea Samorè, University of Bologna, Italy, andrea.samore@unibo.it
Laurent Seppecher, École Centrale de Lyon, France, laurent.seppecher@ec-lyon.fr

Imaging techniques are powerful tools for the analysis of human organs, biological systems or
mechanical and optical materials. They range from different kinds of tomography such as X-ray CT(Computed Tomography), EIT (Electrical Impedance Tomography), DOT (Diffuse Optical
Tomography), PT (Photoacoustic Tomography), OCT (Optical Coherence Tomography), UltrasoundImaging, Elastography, etc. to different kinds of microscopy (fluorescence microscopy, singlemolecule fluorescence, Light Sheet Fluorescence Microscopy, etc.). They are based on the solution of
inverse problems that require mathematical modeling of the acquisition process and numericalmethods for the solution of the equations relating the acquired data to the unknown object.
The purpose of this mini-symposium is to bring together researchers with different backgrounds andinterests in Inverse Problems in Biomedical and Material Imaging. All researchers, academicians,
practitioners, as well as students interested in these knowledge areas, are invited to submit theirrecently achieved results in modeling, solution methods and their use on real data from biomedical
applications and non destructive material testing.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DIRECT INVERSION METHOD FOR QUASI-STATIC MEDICAL ELASTOGRAPHY:STABILITY AND DISCRETIZATION

E. Bretina, P. Millienb, L. SeppechercaInstitut Camille Jordan, Insa-Lyon, Univ Lyon, France, elie.bretin@insa-lyon.fr;bInstitut Langevin, PSL Research University, Paris, France, pierre.millien@espci.fr
cInstitut Camille Jordan, Ecole Centrale de Lyon, Univ Lyon, France, laurent.seppecher@ec-lyon.fr;

Abstract
This work focuses on the reconstruction of shear modulus of biological tissues from elastic internal
displacement fields having in mind some application in quasi-static elastography.
More precisely, we are interested in a direct method, obtained from the reverse weak formulation of thelinear elasticity equation, and where the elastic moduli are found by identifying the nullspace of a linear operator
A constructed directly from the displacement fields.
The resulting method is efficient because it does not require iterative resolution of the forward problem
nor additional information at the boundary. Another advantage is that it has stability properties under relatively
weak regularity assumptions.
After presenting some stability results in a continuous viewpoint (see [1]), we will explain how topreserve its good properties after discretization of the operator A by a finite element approach.
In particular, we will show how to obtain quantitative error estimates (see [2]) for this inverse problem using awell-chosen pair of finite element spaces satisfying a generalized discrete inf-sup condition.
Numerical experiments will illustrate such stability results.

References
1. E. Bretin, P. Millien and L. Seppecher, Stability for finite element discretization of some elliptic inverse
parameter problems from internal data—application to elastography, Siam Journal on imaging Sciences(Accepted).
2. H. Ammri, E. Bretin, P. Millien and L. Seppecher, A direct linear inversion for discontinuous elastic
parameters recovery from internal displacement information only, Numerische Mathematik, 1, 189-226 (2021).

230

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

MAPPING THE RELATIVE SHEAR MODULUS WITHIN BIOLOGICAL TISSUESFROM INTERNAL DISPLACEMENT FIELDS MEASURED IN QUASI-STATICULTRASOUND ELASTOGRAPHY

E. Brusseaua, L. Petruscaa, E. Bretinb, P. Millienc, L. Seppecherda
CREATIS, Univ Lyon, France, elisabeth.brusseau@creatis.insa-lyon.fr; lorena.petrusca@creatis.insa-lyon.frbInstitut Camille Jordan, INSA de Lyon, Univ Lyon, France, elie.bretin@insa-lyon.frcInstitut Langevin, PSL Research University, Paris, France, pierre.millien@espci.frdInstitut Camille Jordan, Ecole Centrale de Lyon, Univ Lyon, France, laurent.seppecher@ec-lyon.fr

Abstract
This presentation deals with relative shear modulus mapping within linear elastic and isotropic media,
from the knowledge of displacement fields estimated in quasi-static ultrasound elastography (QSUE). QSUE
mainly provides images of the internal displacements and strains that biological tissues experience undercompression. Typically, the operator compresses the biological tissues with the probe while radiofrequency (RF)ultrasound images are acquired. With this technique, no boundary information is therefore available. Moreover,internal displacement fields are estimated from the acquired RF images, leading to 2D displacements, whereas
the inverse problem is inherently a 3D one. In our recent work, mathematical developments for analyzing thisinverse elasticity problem were introduced, as well as some initial results [1,2]. Here, to overcome the 2D/3D
data problem, a plane stress-based approach is considered. The proposed method is based on the variational
formulation of the equilibrium equations. This approach involves the use of test functions, chosen in this work tovanish at the domain boundary, because of the lack of information. Using a suitable finite element discretization
technique leads to a finite dimensional homogeneous system of linear equations and the problem ofreconstructing a relative shear modulus map is finally reduced to a null space identification problem of a large
sparse matrix. With this method, only the displacement fields within the analyzed media need to be known
beforehand. They are estimated using a method we previously developed for strain imaging [3].
Relative shear modulus mapping was assessed with plane stress and 3D numerical simulations, as well
as data from phantom experiments (CIRS models 049 and 059). These different media consist of sphericalinclusions embedded in homogeneous background materials. In all cases, the reconstructed maps allow a clear
identification of the regions differing in stiffness, with inclusion-to-background modulus ratios (IBRs) thatunambiguously reveal whether inclusions are stiffer or softer than the surrounding material. More specifically,
for the plane stress simulations, resulting IBRs are very close to the theoretical values, whereas those obtainedwith the 3D simulations are, as expected, less accurate since the plane stress conditions are no longer satisfied.
For the experimental data, a comparison with a clinical ultrasound scanner (Aixplorer, SuperSonic Imagine,
France) was also conducted. Overall, for the phantom cases, the IBRs obtained with our approach are closer tothe values given by the phantom manufacturer than the IBRs from the clinical system. Finally, results obtainedwith in vivo data of breast lesions will also be presented and discussed.

References
1. H. Ammari et al., A direct linear inversion for discontinuous elastic parameters recovery from internal
displacement information only, Numerische Mathematik, 147, 189-226 (2021).
2. E. Brusseau et al., Reconstructing the shear modulus contrast of linear elastic and isotropic media in quasistatic ultrasound elastography, IEEE EMBS, 3171-74 (2021).
3. E. Brusseau et al., In vivo response to compression of 35 breast lesions observed with a two-dimensional
locally regularized strain estimation method, UMB, 40, 300-312 (2014).

231

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RECENT ADVANCES IN CORRELATION-BASED SUPER-REVOLUTION
FLUORESCNT MICROSCOPY: FROM SPARSE/NON-CONVEX TO GENERATIVEAPPROACHES

Luca Calatroni
Laboratoire d'Informatique, Signaux et Systèmes de Sophia-Antipolis (I3S), Sophia Antipolis – France,
calatroni@i3s.unice.fr

Abstract
Super-resolution fluorescence microscopy overcomes the physical barriers due to light diffraction,
allowing for the observation of thin and small entities with sub-pixel precision. State-of-the-art super-resolution
methods achieve adequate spatio-temporal resolution under rather challenging experimental conditions by means
either of costly devices and/or very specific fluorescent molecules.
In this talk, we present a method for covariance-based super-resolution microscopy with intensity
estimation and automatic parameter estimation which is well suited for live-cell imaging and which allows for
improved spatio-temporal resolution by means of common microscopes and conventional fluorescent dyes. Our
approach codifies the assumption of sparse distribution of the fluorescent molecules as well as the temporal andspatial independence between emitters in a covariance domain where the location of emitters is estimated bysolving a non-convex optimisation problem where the $\ell_0$ pseudo-norm is relaxed in an appropriate way. A
separate intensity estimation step where intensity information is retrieved is then considered. This is a valuablepiece of information for the biological interpretation of the results and their use in 3D super-resolution imaging,such as Multi Angle Total Internal Reflectance Fluorescence (MA-TIRF) microscopy. for which results are
showed. To make the reconstruction model fully automated, we further detail automatic parameter selection
strategies based on algorithmic restarting and on discrepancy-type approaches.
To conclude, we discuss how the latent fluctuation model can be estimated in a hybrid fashion
combining sparsity constraints with Generative Adversarial Networks (GANs).
This is joint work with M. Cachia, V. Stergiopoulou, L. Blanc-Féraud (CNRS, I3S, Sophia-Antipolis),
H. Goulart (IRIT, ENSEEIHT, Toulouse) and S. Schaub (IMEV, Villefranche-sur-Mer).

References[1] V. Stergiopoulou, L. Calatroni, H. Goulart, S. Schaub, L. Blanc-Féraud, COL0RME: Super-resolution
microscopy based on sparse blinking fluorophore localization and intensity estimation, to appear in
Biological Imaging, (2022).
[2] V. Stergiopoulou, L. Calatroni, S. Schaub and L. Blanc-Féraud, 3D Image Super-Resolution by fluorophore
fluctuations and MA-TIRF Microscopy reconstruction (3D-COL0RME), to appear in 2022 IEEE 19thInternational Symposium on Biomedical Imaging (ISBI) (2022).
[3] V. Stergiopoulou, H. Goulart, S. Schaub, L. Calatroni and L. Blanc-Féraud, COL0RME: COvariance-based$ell_0$ super-Resolution Microscopy with intensity Estimation, 2021 IEEE 18th International Symposium
on Biomedical Imaging (ISBI) (2021).

232

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A DEEP CONVOLUTIONAL NEURAL NETWORK FOR BRIGHTNESSESTIMATION IN FLUORESCENCE IMAGING

Andrea Samorèa, Damiana LazzarobaDepartment of Mathematics, University of Bologna, Bologna, Italy andrea.samore@unibo.it;b
Department of Mathematics, University of Bologna, Bologna, Italy damiana.lazzaro@unibo.it;

Abstract
Deep learning techniques have recently revolutionized the imaging field, often achieving impressive results
and super-human level performance on complex computer vision tasks such as classification and object
detection. Nonetheless, their use in biomedical imaging is often complicated by the limited amount of available,
high-quality annotated data, which is crucial for training neural networks.
The successful application of a convolutional neural network (CNN) for brightness estimation of fluorescentspots in a point of care embedded system for viral diagnosis is presented [1]. For this problem, object detection is
generally required to assess the presence of a molecule of interest, while brightness estimation is often performed
to measure its concentration [2].
A new neural network for object detection and brightness estimation, which allows for completely automatedfluorescence image analysis without requiring any input parameter from the user during the inference phase, ispresented. Additionally, methods to assemble a good dataset with a limited number of images, as is typical withclinical data, will be discussed, together with strategies to reduce the computational load of the network throughmodel optimization for deployment on resource-constrained embedded systems.

References
1. A. Samorè, M. Rusci, D. Lazzaro, P. Melpignano, L. Benini and S. Morigi, BrightNet: A Deep CNN for
OLED based point of care immunofluorescent diagnostic systems, in IEEE Transactions on Instrumentationand Measurement (2020).
2. Lazzaro, Damiana; Morigi, Serena; Melpignano, P.; Loli Piccolomini, E.; Benini, Luca, Image enhancement
variational methods for Enabling Strong Cost Reduction in OLED-based Point- of-Care Immunofluorescent
Diagnostic Systems, International Journal for Numerical Methods in Biomedical Engineering, 2018, 34, pp.
2932 - 2951

233

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SPATIALLY ADAPTIVE IMAGE RECONSTRUCTION IN ELECTRICALIMPEDENCE TOMOGRAPHY

Serena Morigi
Department of Mathematics, University of Bologna , Bologna, Italy
serena.morigi@unibo.it

Abstract
The inverse Electrical Impedance Tomography (EIT) problem involves collecting electricalmeasurements on the smooth boundary of a region to determine the spatially varying electrical conductivity
distribution within the bounded region. Effective applications of EIT technology emerged in different areas ofengineering, technology and applied sciences. The feasibility studies on potential medical imaging andbiomaterial-engineering applications of EIT have focused on non-invasive approaches to monitoring and data
reconstruction [1]. However, EIT is well known to suffer from a high-degree of nonlinearity and severe
illposedness. Therefore, regularization is required to produce reasonable electrical impedance images. Using
difference imaging, we propose a spatially-variant variational method which couples sparsity regularization andsmoothness regularization for improved data reconstruction. The EIT variational model can benefit fromstructural prior information in the form of an edge detection map coming either from an auxiliary image of thesame object being reconstructed or automatically detected.
We propose an efficient algorithm for minimizing the (non-convex) function based on Alternating
Direction Method of Multipliers (ADMM). Preliminary experiments are presented which strongly indicate thatusing nonconvex versus convex variational EIT models holds the potential for more accurate reconstructions.

References
1. M. Cortesi, A. Samorè, J. Lovecchio, S. Morigi, E. Giordano, EIT for tissue engineering applications: a casefor osteogenic differentiation, 20th International Conference on Biomedical Applications of Electrical
Impedance Tomography (EIT 2019), 1-3 July 2019, London, 2019234

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

HIGH SENSITIVITY IMAGING OF DEFECTS IN ELASTIC WAVEGUIDES USINGNEAR RESONANCE FREQUENCIES
A. Niclasa , E. Bonnetierb, L. Seppecherc and G. VialdaInstitut Camille Jordan, Ecole Centrale de Lyon, Lyon, France, angele.niclas@ec-lyon.frbInstitut Fourier, Université Grenoble Alpes, Grenoble, France, Eric.Bonnetier@univ-grenoble-alpes.frcInstitut Camille Jordan, Ecole Centrale de Lyon, Lyon, France, laurent.seppecher@ec-lyon.frdInstitut Camille Jordan, Ecole Centrale de Lyon, Lyon, France, gregory.vial@ec-lyon.fr

Abstract
Reconstruction of defects in waveguides is of crucial interest in nondestructive evaluation of structures.
This work is based on physical experiments done at Institut Langevin where one tries to reconstruct width
defects in thin elastic plates using multi-frequency surface measurements (see [1]). Contrary to usual
backscattering methods avoiding cut-off frequencies of the plate, they developed an experimental inversion
method using these frequencies to obtain a high sensibility reconstruction of local defects in plates. We try here
to provide a mathematical understanding of this inversion method and theoretical results on its stability.
Near resonance frequencies of the waveguide, the elastic problem is known to be ill-conditioned. Using
the framework developed in [2], we prove that a phenomenon close to the tunnel effect in quantum mechanics
can be observed and locally resonant modes propagate in the waveguide (see [3]). These modes are very
sensitive to width variations, and measuring their amplitude enables reconstructing the local variations of the
waveguide shape with very high sensibility. Given surface wavefield measurements for a range of nearresonance frequencies, we provide a stable numerical reconstruction of the width of a slowly varying waveguideand illustrate it on defects like dilation or compression of a waveguide.

References
1.O. Balogun, T. W. Murray and C. Prada, Simulation and measurement of the optical excitation of the S1 zerogroup velocity lamb wave resonance in plates. Journal of Applied Physics 102 (2007) pp. 064914.
2. V. Pagneux and A. Maurel, Lamb wave propagation in elastic waveguides with variable thickness. Proc. R.
Soc. A 462 (2006), pp. 1315–1339.
3. E. Bonnetier, A. Niclas, L. Seppecher and G. Vial, The Helmholtz problem in slowly varying waveguides at
locally resonant frequencies. Submitted in Wave Motion (2022).

235

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

RISING: A NEW FRAMEWORK FOR FEW-VIEW TOMOGRAPHIC IMAGE
RECONSTRUCTION WITH DEEP LEARNING
E. Loli Piccolominia , D. Evangelistab and E. MorotticaDepartment of Computer Science and Engineering, University of Bologna, Italy, elena.loli@unibo.it;b
Department of Mathematics, University of Bologna, Italy, davide.evangelista3@unibo.it;c
Department of Political and Social Sciences, University of Bologna, Italy , elena.morotti4@unibo.it;

Abstract
In this talk we propose a new two-step procedure for sparse-view tomographic image reconstruction. It iscalled RISING, since it combines an early-stopped Rapid Iterative Solver with a subsequent Iteration Networkbased Gaining step.
So far, regularized iterative methods have widely been used for X -ray computed tomography image
reconstruction from low-sampled data, since they converge to a sparse solution in a suitable domain, as upheldby the Compressed Sensing theory. Unfortunately, their use is practically limited by their high computational
cost which imposes to perform only a few iterations in the available time for clinical exams. Data-driven
methods, using neural networks to post-process a coarse and noisy image obtained from geometrical algorithms,
have been recently studied and appreciated for both their computational speed and accurate reconstructions.
However, there is no evidence, neither theoretically nor numerically, that neural networks based algorithms solve
the mathematical inverse problem modeling the tomographic reconstruction process.
In our two-step approach, the first phase executes very few iterations of a regularized model-based algorithm
whereas the second step completes the missing iterations by means of a neural network. The resulting hybrid
deep-variational framework preserves the convergence properties of the iterative method and, at the same time, it
exploits the computational speed and flexibility of a data-driven approach.
Experiments performed on a simulated and a real data set confirm the numerical and visual accuracy of thereconstructed RISING images in short computational times.

236

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A SCALED ADAPTIVE FISTA-LIKE ALGORITHM FOR SUPER-RESOLUTIONIMAGE MICROSCOPY
S. Rebegoldia, L. Calatronib, C. Estaticoc, M. Lazzarettica
Dipartimento di Ingegneria Industriale, Università di Firenze, Via di S. Marta 3, 50139 Florence, Italysimone.rebegoldi@unifi.it;b
CNRS, UCA, Inria, Lab. I3S, 2000 Route des Lucioles, 06903 Sophia-Antipolis, France, calatroni@i3s.unice.fr;cDipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genoa, Italy,
estatico@dima.unige.it; marta.lazzarettu@edu.unige.it;

Abstract
In this talk we propose SAGE-FISTA [1], a Scaled Adaptive GEneralized FISTA-type algorithm for theminimization of the sum of two (possibly strongly) convex functions, one of which is assumed to becontinuously differentiable. SAGE-FISTA can be considered as an accelerated version of the popular FISTA
algorithm [2], where each iterate is obtained by first extrapolating the two previous iterates and then performinga proximal-gradient step at the extrapolated point. Unlike FISTA, the proposed algorithm employs an adaptive
backtracking procedure, which allows for the non-monotone adjustment of the steplength along the iterations.
This comes in handy when the initial guess for the steplength is extremely small, due to a coarse large estimateof the Lipschitz constant. The adaptive backtracking is further combined with a symmetric, positive definite andvariable scaling matrix in the computation of the proximal-gradient step, by which some second-orderinformation of the differentiable part of the objective function are incorporated while keeping the iteration
computationally inexpensive, e.g. adopting Majorization-Minimization or Split-Gradient Techniques [3]. The
algorithm also takes into account the inexact evaluation of the proximal operator and the possible introduction of
the strong convexity moduli of the two functions into the inertial parameters.
If one or both functions are strongly convex, we prove that SAGE-FISTA converges linearly in the function
values, with a convergence factor depending on both the strong convexity moduli of the two functions and theupper and lower bounds on the eigenvalues of the scaling matrices. Otherwise, we show that SAGE-FISTA
retains the same O(1/k^2) convergence rate available for FISTA.
Furthermore, we apply our algorithm to an image super-resolution problem where a sparsity-promoting
regularization function is coupled with a weighted least squares term [4]. Our numerical experiments show that
SAGE-FISTA boosts the practical convergence speed with respect to standard implementations of FISTA,
especially when used as inner solver of iteratively reweighted ℓ1 schemes.

References
1. S. Rebegoldi and L. Calatroni, Scaled, inexact and adaptive generalized FISTA for strongly convex
optimization, https://arxiv.org/abs/2101.03915 (2021).
2. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM
J. Imaging Sci. 2, 183-202 (2009).
3. S. Bonettini, S. Rebegoldi and V. Ruggiero, Inertial variable metric techniques for the inexact forwardbackward algorithm, SIAM J. Sci. Comput. 40(5), A3180-A3210 (2018).
4. M. Lazzaretti, S. Rebegoldi, L. Calatroni and C. Estatico, A scaled and adaptive FISTA algorithm for signaldependent sparse image super-resolution problems, Lect. Notes Comput. Sci. 12679 (2021).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A VARIATIONAL APPROACH FOR JOINT IMAGE RECOVERY-SEGMENTATIONBASED ON SPATIALLY VARYING GENERALISED GAUSSIAN MODELS

É. Chouzenoux a, M.-C. Corbineaua,b, J.-C. Pesqueta, G. ScrivantiaaUniversité Paris-Saclay, Inria, CentraleSupélec, CVN, 9 rue Juliot Curie, 91190, Gif-sur-Yvette, France,
firstname.lastname@centralesupelec.fr;bResearch Department Preligens, 49/51 Rue de Provence, 75009, Paris, France,
mariecaroline_corbineau@hotmail.fr;

Abstract
The joint problem of reconstruction/feature extraction is a challenging task in image processing. It consists inperforming, in a joint manner, the restoration of an image and the extraction of its features. The degradation
usually corresponds to the application of a linear operator (e.g., blur, projection matrix) to the image, and the
addition of a noise. Feature extraction problems arise when one wants to assign to the sought image a small setof parameters which can describe or identify the image itself. Image segmentation can be viewed as an example
of features extraction, which consists of defining a label field on the image domain so that pixels are partitioned
into a predefined number of homogeneous regions according to some specific characteristics.
In this work, we firstly propose a novel non-smooth and non-convex variational formulation of the joint
problem of reconstruction/feature extraction. To do so, we consider the notion of flexible sparse regularization,
introduced in [1]: for some, this amounts to considering penalties of the form,
with positive exponents, for the variational regularisation of ill-posed inverse problems. To include thisclass of penalties in our formulation, we introduce a versatile prior that is based on a mixture of generalised
Gaussian distributions (GGD) and whose parameters, including its exponents, are space-variant. We thus aim at
jointly estimating an optimal configuration for, and retrieving the image. Under an assumption of
consistency within the exponents values of a given region of the features space, we obtain the desired feature
extraction starting from the estimated parameters.
The specific non-convex structure of the resulting objective function, which is the sum of a coupling term
and a set of block-separable terms, suggests the use of an alternating scheme: the idea is to sequentially update asubset of parameters through the resolution of an inner minimization problem, while the other parameters are
assumed to be fixed. We design a new proximal alternating minimization algorithm that efficiently exploits thecharacteristic of the functions involved in our proposed variational formulation. We consider a hybrid scheme
mixing both standard and linearised proximal steps on the different blocks of parameters. The novelty lies in thefact that the linearised step exploits a preconditioner for faster convergence and, contrarily to what usually
happens in other methods belonging to this family, the linearisation involves a block-separable term, instead of
the coupling term. We investigate the convergence properties for this algorithm based on the framework definedin [2].
Finally, we test the proposed approach in a joint deblurring/segmentation task for realistically simulated
ultrasound images and compare its performance with state-of-the-art method in the field [3].

References
1. D. A. Lorenz & E. Resmerita,. Flexible sparse regularisation, Inverse Problems, 33, (2017).
2. H. Attouch, J. Bolte & B.F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems:
proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods, Mathematical
Programming, 137, 91–129, (2013).
3. M.-C. Corbineau, D. Kouamé, É. Chouzenoux, J. Tourneret, & J.-C. Pesquet, Preconditioned P-ULA for Joint
Deconvolution-Segmentation of Ultrasound Images. IEEE Signal Processing Letters, 26, 1456-1460. (2019).

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

OPTIMAL REGULARIZED ESTIMATION OF THE CROSS-POWER SPECTRUM
FROM INDIRECT MEASUREMENTS: THEORETICAL RESULTS ANDAPPLICATION TO BRAIN CONNECTIVITY
S. Sommarivaa , E. Vallarinoa, M. Pianaa,b and A. Sorrentinoa,ba
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genoa, Italy,
sommariva@dima.unige.it , vallarino@dima.unige.it , piana@dima.unige.it , sorrentino@dima.unige.it;b
CNR-SPIN, Corso Perrone 24, 16152, Genoa, Italy.

Abstract
We consider the problem of estimating the cross-power spectrum of a multivariate stochastic process
when only indirect measurement are available through a second observable stochastic process. We assume
the two processes to be related by a linear model of the form

(1)
A typical approach to this problem consists in two steps. First and estimate ofis obtained from therecorded data by solving the inverse problem described by equation (1). Then the cross-power spectrum of suchan estimate is computed.
In [1] and [2] we have shown, analitically and through simulations, respectively, that if the inverse problem issolved via Tikhonov regularization, an unexpected parameter tuning issue may rise. Indeed the value of theregularization parameter that provides the best reconstruction of the unknown(i.e. minimum -norm of
the reconstruction error) is in general suboptimal for the subsequent estimation of the cross-power spectrum.
First, I will present the theoretical results we obtained by assuming both the signal
and the noise
to be white Gaussian noise. Then, I will show, through numerical simulations, that the analytical results hold also
in case of more complex signals. Finally, I will indtroduce some preliminary results on the potentials of a onestep approach to the problem, i.e when the cross-power spectrum ofis directly estimated from that of. At the end, I will discuss the implication of such results when studying functional brain connectivity frommagneto-/electro-encephalographic data, i.e. when estimating the statistical relationship between the activity ofspatially distinct brain areas from the electromagnetic field non invasively recorded outside the scalp [3, 4].

References
1. E. Vallarino, S. Sommariva, M. Piana, A. Sorrentino. On the two–step estimation of the cross–power spectrum for dynamical inverse problems, Inverse Problems, 36 (4), 045010 (2020).
2. Vallarino, E., Sorrentino, A., Piana, M., & Sommariva, S. The role of spectral complexity in connectivity
estimation. Axioms, 10(1), 35 (2021).
3. S. Supek, and J. A. Cheryl, eds. Magnetoencephalography: From Signals to Dynamic Cortical Networks.
Springer, 2019.
4. S. Sommariva, A. Sorrentino, M. Piana, V. Pizzella and L. Marzetti, A comparative study of the robustness of
frequency–domain connectivity measures to finite data length, Brain Topography, 32, 675-695 (2019).

239

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

FREE HYPER-PARAMETER SELECTION AND AVERAGING IN
MAGNETO/ELECTRO-ENCEPHALOGRAPHY INVERSE PROBLEM
Alessandro Viania, Alberto SorrentinobaDepartment of Mathematics, University of Genova, Genova, Italy viani@dima.unige.it;b
Department of Mathematics, University of Genova, Genova, Italy sorrentino@dima.unige.it;

Abstract
We present an innovative method for exploiting Sequential Monte Carlo (SMC) samplers structure in orderto obtain free hyper-parameter selection and averaging for a large class of inverse problems where the likelihood
depends on a scalar hyper-parameter for which an hyper-prior is available.
Indeed, considering a sequence of intermediate SMC samplers distributions such that each one can beinterpreted as a posterior distribution conditioned on a different value for the hyper-parameter, we can exploit thenatural approximation of the normalizing constant provided by the algorithm to obtain either selection or
marginalization of the hyper-parameter.
The proposed approach is substantially different from the standard alternative where the hyper-parameter isitself sampled and estimated directly by the SMC samplers because particles from all iterations are effectively
utilized in the final estimate, therefore avoiding the waste of computational time that is typical of SMC samplers.
In order to give an example for the construction of such sequences, we focus on a class of inverse problems
where the likelihood belongs to the Natural Exponential Family (NEF), showing that in this context the mostnatural choice for the sequence of distributions - the one obtained growing the likelihood to an increasing
exponent from zero to one - is suitable for the proposed method.
The most straightforward application is in the context of additive Gaussian noise inverse problems, where thenoise standard deviation plays the role of the hyper-parameter, such as in the field of source localization from
Magneto/Electro-Encephalography (MEG) data [1,2].
For analyzing the performance of the proposed method, we provide a comparison between the performancesof the proposed method and the standard alternative for the resolution of both a toy example and the MEG
inverse problem, showing that the proposed method provides better results with less computational cost.

References
1. Sommariva, Sorrentino (2014). Sequential monte carlo samplers for semi-linear inverse problems andapplication to magnetoencephalography. Inverse Problems 30(11), 114020.
2. Viani, Luria, Bornfleth, Sorrentino (2021, oct). Where bayes tweaks gauss: Conditionally gaussian priors forstable multi-dipole estimation. Inverse Problems and Imaging 15(5), 1099–1119.

240

MINISYMPOSIUM
M26: Variational Methods for Inverse Problems in Imaging
Organizers:
Robert Beinert, Technische Universität Berlin, Germany, robert.beinert@tu-berlin.de
Kristian Bredies, University of Graz, Austria, kristian.bredies@uni-graz.at
Variational methods nowadays rank among the most powerful and flexible approaches totackle inverse problems in imaging. These include real-world applications like magnetic resonanceimaging, positron emission tomography, computer tomography, transmission electron microscopy andmany other recovery problems in medicine, engineering, and life sciences. Further, variational
methods have proven themselves highly valuable for data pre-processing and image post-processing
such as denoising, deblurring and inpainting. The recent years have seen several new developments inthis area, coming from different mathematical backgrounds and being applicable for a variety of
inverse problems. These developments include, for instance, deep learning approaches, tensorial lifting
strategies, as well as novel optimal-transport- and total-variation-based regularizers. On the other
hand, new techniques such as deep neural networks for inverse problems are inspired by variational
methods. The aim of this minisymposium is to bring together experts with various backgrounds todiscuss these recent achievements in the context of inverse problems in imaging, and to initiate
potential new research directions and collaborations.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

ROBUST PCA VIA REGULARIZED REAPER AND MATRIX-FREE PROXIMALALGORITHMS

R. Beinerta, G. SteidlbaTechnische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany,robert.beinert@tu-berlin.debTechnische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany,steidl@math.tu-berlin.de

Abstract
The Principal Component Analysis (PCA) is a dimension reduction technique, where the dimension of agiven data set is reduced by projecting them to a low-dimensional, affine subspace. Classically, the subspace
which minimizes the sum of the squared Euclidean distances between the data points and their projections ischosen. This PCA approach is however very sensitive to outliers, so that various robust approaches weredeveloped in the literature.
A recent robust model, called REAPER [1], aims to find the principal components by solving a convex
optimization problem. Usually, the number of principal components must be determined in advance and theminimization is performed over symmetric positive semi-definite matrices having the size of the data, although
the number of principal components is substantially smaller. This prohibits its use if the dimension of the data islarge, which is often the case in image processing.
In this talk, we propose a regularized version of REAPER, called rREAPER [2], which enforces the sparsityof the number of principal components by penalizing the nuclear norm of the corresponding orthogonal
projector. If only an upper bound on the number of principal components is available, our approach can be
combined with the L-curve method to reconstruct the appropriate subspace. To solve the variational problem
numerically, we introduce a matrix-free algorithm to find a minimizer of rREAPER which is also suited for highdimensional data. The algorithm couples a primal-dual minimization approach [3] with a thick-restarted Lanczos
process [4]. This appears to be the first efficient convex variational method for robust PCA that can handle highdimensional data. As a side result, we discuss the topic of the bias in robust PCA.

References
1. G. Lerman, M. McCoy, J.A. Tropp, and T. Zhang, Robust computation of linear models by convex relaxation,
Foundations of Computational Mathematics, 15, 363–410 (2015).
2. R. Beinert and G. Steidl, Robust PCA via regularized REAPER with a matrix-free proximal algorithm,
Journal of Mathematical Imaging and Vision, 63, 626-649 (2021).
3. A. Chambolle, T. Pock, An introduction to continuous optimization for imaging, Acta Numerica 25, 161–319(2016).
4. K. Wu, H. Simon, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM Journal on
Matrix Analysis and Applications, 22, 602–616 (2000).

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

JOINT EXIT WAVE RECONSTRUCTION AND IMAGE REGISTRATION AS A
LEAST-SQUARES PROBLEM
Benjamin Berkelsa,b and Christian Dobersteinba IGPM, RWTH Aachen University, Aachen, Germany, berkels@aices.rwth-aachen.de;b AICES, RWTH Aachen University, Aachen, Germany, doberstein@aices.rwth-aachen.de;

Abstract
Images generated by a transmission electron microscope (TEM) are blurred by aberrations from theobjective lens and can be difficult to interpret correctly. One possible solution to this problem is to reconstruct
the so-called exit wave, i.e. the electron wave in the microscope right before it passes the objective lens, from aseries of TEM images acquired with varying focus. The forward model of simulating a TEM image from agiven exit wave is known and easy to evaluate. In fact, it can be represented as weighted auto-correlation of theexit wave  with the transmission cross-coefficient (TCC) T . More specifically,  T  is the Fourier
space representation of the corresponding TEM image. Here, the TCC encapsulates all assumptions on theelectron microscope and the observed sample. However, it is in general not possible to reconstruct the exit wave
from a series of images analytically. Here, it is important to note that the exit wave is complex-valued, while theTEM images are only real-valued. The corresponding inverse problem can be formulated as a minimization
problem, which is done in the well known multiple input maximum a-posteriori (MIMAP) algorithm [1] and the
maximum-likelihood (MAL) algorithm [2].
We propose a generalization of these methods by performing the exit wave reconstruction and theregistration of the image series simultaneously using the functional

E : L2 ( A, C)  ( R 2 ) N  R ,, t   1 Nj1  TNZj  F ( g j  t j )22L    M

2
L2

.

Here, g1 , ..., g N are TEM images corresponding to focus values Z1 , ..., Z N , t is a translation by thejvector t j and M is a fixed a-priori estimate of the exit wave. We show that the objective functional is notconvex with respect to the exit wave for   0 sufficently small, which also carries over to the MAL and
MIMAP functionals.
The main result is the existence of minimizers of the objective functional. These results are based on theproperties the weighted cross-correlation and certain assumptions on the TCC. A key ingredient here is theuniform approximation of the TCC with a sum of  -seprable weights.
Finally, the practical applicability of our method is verified with numerical experiments on simulated
and real input data.
Note that part of this work has been published in [3].

References
1. E. J. Kirkland. Improved High Resolution Image Processing of Bright Field Electron Micrographs,
Ultramicroscopy, 15, 151-172 (1984).
2. W. Coene, A. Thust, M. Op de Beeck, and D. Van Dyck. Maximum-Likelihood Method for Focus-Variation
Image Reconstruction in High Resolution Transmission Electron Microscopy, Ultramicroscopy, 64, 109-135(1996).
3. C. Doberstein and B. Berkels. A least-squares functional for joint exit wave reconstruction and image
registration, Inverse Problems, 35 (2019).

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

OPTIMAL-TRANSPORT-BASED APPROACHES FOR DYNAMIC IMAGE
RECONSTRUCTION

Kristian Brediesa, Marcello Carionib, Silvio Fanzonc and Francisco Romeroda
University of Graz, Heinrichstraße 36, 8010 Graz, Austria, kristian.bredies@uni-graz.atb
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK, mc2250@maths.cam.ac.uk
cUniversity of Graz, Heinrichstraße 36, 8010 Graz, Austria, silvio.fanzon@uni-graz.atdUniversity of Graz, Heinrichstraße 36, 8010 Graz, Austria, francisco.romero-hinrichsen@uni-graz.at

Abstract
We discuss the solution of dynamic inverse problems in which for each time point, a time-dependent linear
forward operator mapping the space of measures to a time-dependent Hilbert space has to be inverted. These
problems are regularized with dynamic optimal-transport energies that base on the continuity equation as well as
convex functionals of Benamou-Brenier-type. Well-posedness of respective Tikhonov minimization is discussed
in detail. Further, for the purpose of deriving properties of the solutions as well as numerical algorithms, we
present sparsity results for general inverse problems that are connected with the extremal points of the BenamouBrenier energy subject to the continuity equation. For the latter, it is proven that the extremal points are realized
by point masses moving along curves with Sobolev regularity. This result will be employed in numerical
optimization algorithms of generalized conditional gradient type. We present instances of this algorithm that are
tailored towards dynamic inverse problems associated with point tracking. Finally, the application and numericalperformance of the method is demonstrated for sparse dynamic superresolution.

References
1. Kristian Bredies, Silvio Fanzon. An optimal transport approach for solving dynamic inverse problems inspaces of measures. ESAIM: Mathematical Modelling and Numerical Analysis, 54(6):2351-2382, 2020.
2. Kristian Bredies, Marcello Carioni. Sparsity of solutions for variational inverse problems with finitedimensional data. Calculus of Variations and Partial Differential Equations, 59:14, 2020.
3. Kristian Bredies, Marcello Carioni, Silvio Fanzon, Francisco Romero. On the extremal points of the ball of the
Benamou-Brenier energy. Bulletin of the London Mathematical Society, 53(5):1436-1452, 2021.
4. Kristian Bredies, Marcello Carioni, Silvio Fanzon and Francisco Romero. A generalized conditional gradientmethod for dynamic inverse problems with optimal transport regularization. Foundations of Computational
Mathematics, in press, 2022.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

DEEP NEURAL NETWORKS FOR INVERSE PROBLEMS WITHPSEUDODIFFERENTIAL OPERATORS: AN APPLICATION TO LIMITED ANGLETOMOGRAPHY

Tatiana Bubba
Department of Mathematical Sciences, University of Bath, UKtab73@bath.ac.uk

Abstract
Computed Tomography makes use of computer-processed combinations of many X-ray measurementsof an object, taken from different angles, and attempts to recover the inner structure of the object from the data.
In the case of limited-angle tomography, the reconstruction problem is severely ill-posed and the traditional
reconstruction methods, e.g. filtered backprojection (FBP), do not perform well. In this work, we investigate abrand-new method for limited-angle tomography reconstruction, based on the unrolled version of the ISTA
algorithm where each iteration contains a convolutional neural network (CNN). The idea of this project has
emerged from the observation that the backprojection operator can be approximated by a sequence ofconvolutions applied to the original object in the wavelet domain. Thus, each CNN has the same structure as thepreviously mentioned sequence of convolutions. The preliminary results on 128x128 images, highly
encouraging, show that our model can provide reconstructions that are much more accurate than thereconstructions provided by the standard ISTA algorithm.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

TOWARDS OFF-THE-GRID ALGORITHMS FOR TOTAL VARIATIONREGULARIZED INVERSE PROBLEMS

Y. De Castroa, V. Duvalb,c, R. Petitc,baInstitut Camille Jordan, CNRS UMR 5208, École Centrale de Lyon, F-69134 Écully, France yohann.decastro@ec-lyon.fr;bInria Paris, MOKAPLAN, 75012 Paris, France vincent.duval@inria.fr;c
CEREMADE, CNRS UMR 7534, Unversité Paris-Dauphine, PSL University, 75016 Paris, France,romain.petit@inria.fr;

Abstract
The choice of a regularizer has a crucial impact on the structure of the solutions of an inverse problem. It is
now well understood which regularizations tend to promote sparse vectors, low rank matrices, or piecewise
constant images. Yet, such structural properties are rarely exploited in optimization algorithms. In most cases,
they only appear in the limit, when the algorithm has converged.
In this work, we focus on the total (gradient) variation which has been widely used in imaging following thepioneering work of Rudin, Osher and Fatemi [1]. Such a regularization is used to recover piecewise constant(“cartoon”) solutions. Indeed, it is known that some solutions are the sum of a few indicator functions [2,3].
Inspired by recent works on the recovery of point sources [4], we propose to exploit that structure in a FrankWolfe based algorithm which progressively adds shapes which are the solution a variational geometric problem
which is similar to the Cheeger problem.
The benefit of this approach is that the reconstructed signal does rely on a computation grid, therefore it does
not suffer from traditional reconstruction artifacts such as anisotropy or blur. A preprint is available at [5].

References
1. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D:
Nonlinear Phenomena, 60(1):259-268 (1992).
2. K. Bredies and M. Carioni, Sparsity of solutions for variational inverse problems with finite-dimensional data.
Calculus of Variations and Partial Differential Equations, 59(1):14 (2019).
3. C. Boyer, A. Chambolle, Y. De Castro, V. Duval, F. de Gournay and P. Weiss, On Representer Theorems andConvex Regularization, SIAM Journal on Optimization, 29(2):1260-1281 (2019).
4. K. Bredies and H. Pikkarainen, Inverse problems in spaces of measures. ESAIM: Control, Optimization andCalculus of Variations, 19(1):190-218 (2013).
5. Y. De Castro, V. Duval and R. Petit, Towards off-the-grid algorithms for total variation regularized inverse
problems. arXiv preprint arXiv:2104.06706 (2021).

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

PIXEL-DRIVEN TOMOGRAPHY: ANALYSIS AND THE GRATOPY TOOLBOX

R.M. Hubera, K. BrediesbaUniversity of Graz, Heinrichstraße 36, 8010 Graz, Austria, richard.huber@uni-graz.atb
University of Graz, Heinrichstraße 36, 8010 Graz, Austria, kristian.bredies@uni-graz.at

Abstract
The Radon transform is a cornerstone of countless imaging applications associated with tomography and istherefore often considered in the context of inverse problems [1]. The use of computers in tomographic
reconstruction requires discretization of the Radon transform and its adjoint – the backprojection. However, thedesign of discretizations is neither trivial nor obvious. Many applications use different discretization approaches
for the Radon transform and the backprojection, which can have a negative impact on the convergence of
iterative solvers. Moreover, using discretizations of operators – and not the infinite-dimensional operator – in thecontext of variational methods for inverse problem can be understood as the usage of an inexact operator. A
complete analysis of a variational approach has to take such operator errors into account; cf. [2]. Hence, one
needs to know this error – in the best case in a uniform sense – but often, this is ignored. A well-known
discretization approach to the Radon transform – and more general projection methods – is the ‘Pixel-Driven’
method [3], which is typically used for the backprojection but anecdotally known to be a poor approximation for
the forward operator, showing distinct oscillation artifacts [4]. The word ‘anecdotal’ reveals the fact that there isvery little rigorous mathematical analysis of such methods. We present such an analysis for Pixel-Driven
methods, showing convergence in the operator norm towards the Radon transform between L2 spaces when thediscretization parameters are chosen appropriately [5]. Note that while with such parameters, convergence –including rates – can be obtained, this suitable parameter choice does not coincide with the standard choice, thus
explaining the anecdotal poor approximation quality. We further present the Gratopy toolbox [6], a PythonOpenCL-based toolbox allowing for highly efficient, versatile, and easy application of Pixel-Driven methods tomake use of said theoretical results.

References
1. J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances, WA: SPIE The
International Society for Optical Engineering, 2009.
2. S. Lu, S. Pereverzyev, Y. Shao, A. Tautenhahn. . On the generalized discrepancy principle for Tikhonov
regularization in Hilbert scales, Journal of Integral Equations and Applications, 22 (2010).
3. Z. Qiao, G. Redler, Z. Gui, Y. Qian, B. Epel, and H. Halpern, Three novel accurate pixel-driven projection
methods for 2D CT and 3D EPR imaging, Journal of X-ray science and technology, 26, 83-102 (2017).
4. B. De Man and S. Basu, Distance-driven projection and backprojection. 2002 IEEE Nuclear Science
Symposium Conference Record, 3, 1477–1480 (2002).
5. Kristian Bredies and Richard Huber, Convergence analysis of pixel-driven Radon and fanbeam transforms,
SIAM Journal on Numerical Analysis, 59, 1399–1432 (2021).
6. Kristian Bredies and Richard Huber, Gratopy 0.1 [Software]. Zenodo, https://doi.org/10.5281/zenodo.5221442(2021).

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

STEKLOV AND MODIFIED TRANSMISSION EIGENVALUES AS TARGET
SIGNATURES IN AN INVERSE FLUID-SOLID INTERACTION PROBLEM
B. Komandera , D. A. Lorenzb and L. VestwebercaInstitute for Mathematical Optimization, TU Braunschweig, Universitätsplatz 2, 38106
Braunschweig,Germany, b.komander@tu-braunschweig.de;b
Institute of Analysis and Algebra, TU Braunschweig, Universitätsplatz 2, 38106 Braunschweig,
Germany, d.lorenz@tu-braunschweig.decInstitute for Numerical Analysis, TU Braunschweig, Universitätsplatz 2, 38106 Braunschweig,
Germany, l.vestweber@tu-braunschweig.de

Abstract
We revisit total variation denoising and study an augmented model where we assume that an estimate of
the image gradient is available. We show that this increases the image reconstruction quality and derive that theresulting model resembles the total generalized variation denoising method, thus providing a new motivation forthis model. Further, we propose to use a constraint denoising model and develop a variational denoising modelthat is basically parameter free, i.e., all model parameters are estimated directly from the noisy image. Moreover,
we use Chambolle–Pock’s primal dual method as well as the Douglas–Rachford method for the new models. For
the latter one has to solve large discretizations of partial differential equations. We propose to do this in an
inexact manner using the preconditioned conjugate gradients method and derive preconditioners for this.
Numerical experiments show that the resulting method has good denoising properties and also that
preconditioning does increase convergence speed significantly.
Finally, we analyze the duality gap of different formulations of the TGV denoising problem and derivea simple stopping criterion.
This is joint work with M. Levitin (University of Reading) and P. Monk (University of Delaware).

References
1.

B. Komander, D.A. Lorenz and L.
https://doi.org/10.1007/s10851-018-0819-8.

Vestweber,

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Imaging

Vis.

61:

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(2019).

The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

PENALIZED METHODS FOR SOLVING CONSTRAINED VARIATIONAL
PROBLEMS IN IMAGE RECOVERY
Segolene Martina and Jean-Christopher Pesquetba CVN OPIS, 3 rue Jolio Curie, 91190, Gif-sur-Yvette, France, segolene.martin@centralesupelec.fr;bUniversity Paris-Saclay, France, jean-christophe.pesquet@centralesupelec.fr

Abstract
Many image recovery problems can be efficiently addressed by minimizing a smooth cost function
subject to various constraints. In this talk, we present a new method for solving large-scale constrained
differentiable optimization problems. To account efficiently for a wide range of constraints, our approachembeds a subspace algorithm in an exterior penalty framework. The subspace strategy, combined with a
Majoration-Minimization step search, takes great advantage of the smoothness of the penalized cost function.
Assuming that the latter is convex, the convergence of our algorithm to a solution of the constrained
optimization problem is proved.
Numerical experiments carried out on a large-scale image restoration application show that the
proposed method outperforms state-of-the-art algorithms in terms of computational time.

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

VARIATIONAL METHODS FOR FUNCTIONAL AND QUANTITATIVE MRI
R. Stollbergera,b, SM. Spanna, O. MaieraaInstitute of Medical Engineering, Graz University of Technology, Stremayrgasse 16, 8010, Graz, Austria,
rudolf.stollberger@tugraz.atb
BioTechMed-Graz, Graz, Austria

Abstract
Inverse problems are an essential part of functional and quantitative procedures in MRI. These MRI
methods typically require multiple measurements of specific regions for the intended analysis. In this context,
variational reconstruction techniques can contribute considerably to an increased spatial-temporal resolution withintegrated denoising and sub-sampling artifact suppression. The determination of biophysical parameters istypically a separate inverse problem, but can also be integrated into a model-based reconstruction where thephysical model connects the different sample volumes. Here we show how special problem-adapted
regularization functions enable significant progress in dynamic MR imaging, brain perfusion determination by
ALS and for the determination of biophysical parameters.
An essential aspect of regularization for functional and quantitative methods is the combination of regularization
terms for individual components of the information base. Thus, with the infimal convolution of total generalized
variation (ICTGV) a flexible regularization can be achieved, which spatially models fast and slow signal changes[1]. For arterial spin labeling (ASL), two different image data sets have to be measured and combined, for which
a combined regularization model has proven to be particularly suitable [2, 3]. Simulated data were used to
optimize regularization ("data driven learning"). It was observed that the parameter sets found were quite stable
for realistic data variations. For parameter quantification, both in model-based reconstruction and in image
space, the regularization functionals are joined using a Frobenius norm to exploit shared features between theparameters of interest. Jointly regularizing preserves fine structures which could be otherwise wrongly classified
as noise [4, 5].
The impact and potential for improvements of these regularization strategies together with parallel
imaging and tailored sub-sampling strategies will be shown for cutting edge problems. Furthermore theadvantages and challenges compared to the recently implemented deep learning strategies will be discussed.

References
1. Schloegl, M., Holler, M., Schwarzl, A., Bredies, K., Stollberger, R., 2017. Infimal convolution of total
generalized variation functionals for dyn. MRI. Magn. Reson. Med. 78, 142–155
2. Spann, S.M., Kazimierski, K.S., Aigner, C.S., Kraiger, M., Bredies, K., Stollberger, R., 2017. Spatio-temporal
TGV denoising for ASL perfusion imaging. Neuroimage 157, 81–96.
3. Spann M, Shao X, Danny JJ. Wang D JJ. et al. Robust single-shot acquisition of high resolution whole brain
ASL images by combining time-dependent 2D CAPIRINHA sampling with spatio-temporal TGV
reconstruction. Neuroimage 206, doi.org/10.1016/j.neuroimage.2019.116337 (2020)
4. Maier O, Schoormans J, Schloegl M, et al. Rapid T1 quantification from high resolution 3D data with modelbased reconstruction. Magn Reson Med. 2019 Mar;81(3):2072-2089. doi:10.1002/mrm.27502.
5. Maier O, Spann SM, Pinter D et al, Robust Perfusion Parameter Quantification from 3D Single-Shot MultiDelay ASL measurements, ISMRM 2020, Sydney

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The 10th International Conference ”Inverse Problems: Modeling and Simulation” (IPMS-2022)
Malta, May 22 - 28, 2022

ACCELERATING THE SOLUTION OF SPARSE DYNAMIC INVERSE PROBLEMSUSING TOOLS FROM DYNAMIC PROGRAMMING

Vincent Duval and Robert Tovey
MOKAPLAN, INRIA, Paris, France, vincent.duval@inria.fr & robert.tovey@inria.fr

Abstract
Recent work has shown that sparse dynamical inverse problems, such as particle tracking, can bereformulated into a convex variational optimization problem over the space of measures. The resulting problem
resembles a weighted Lasso problem for sparsity combined with an optimal transport regularization to enforce
regularity in time. Early numerical results using a Frank-Wolfe algorithm are promising, we focus on improving
efficiency for larger scaled problems.
Frank-Wolfe is a greedy method which at each iteration approximates the best single path to fit the remaining
data error. This sub-problem is the main computational challenge as it is very high dimensional and non-convex.
We observe that discretizing this ‘best path’ problem allows it to be reformulated as a classical shortest path
problem on a directed acyclic graph. Leveraging this observation enables us to produce much more efficientalgorithms, either using uniform or random discretizations of the sub-problem.
Our approach is validated analytically and numerically. We first prove that our stochastic Frank-Wolfe
algorithm converges to the globally optimal energy almost-surely. The numerical reconstructions are much fasterthan previous methods and we compare the performance of random and uniform discretizations.

251

MINISYMPOSIUM
M27: Inverse Problems in Scattering Theory and Geometry
Organizers:
Matti Lassas, University of Helsinki, Finland, matti.lassas@helsinki.fi
Teemu Tyni, University of Helsinki, Finland, teemu.tyni@helsinki.fi
Inverse scattering problems and geometrical inverse problems are classical and active subfields of
inverse problems. The aim in inverse scattering theory is to recover information about some unknown
medium or potential function from measurements conducted far away. Correspondingly, often ingeometrical inverse problems the measurement is conducted at the boundary of a region of interest.
Both fields have recently seen many breakthroughs, especially in the non-linear setting.
This mini-symposium aims to collect both experts and newcomers from different fields studying
scattering theory and geometrical inverse problems. Both theoretical and numerical sides of the
problems are welcome.

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The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

DETERMINING A LORENTZIAN METRIC FROM THE SOURCE-TO-SOLUTION
MAP FOR THE RELATIVISTIC BOLTZMANN EQUATION

T. Balehowskya, A. Kujanpääb, M. Lassasb, T. LiimatainenbaUniveristy of Calgary, Mathematical Sciences 476, 2500 University Drive NW, Calgary, Alberta, Canada, T2N
1N4 tracey.balehowsky@ucalgary.cab
University of Helsinki, P.O Box 68 (Pietari Kalmin katu 5), 00014 Helsingin Yliopisto, Helsinki, Finland

Abstract
In this talk, we consider the following problem: Given the source-to-solution map for a relativistic Boltzmann
equation on a neighbourhood V of an observer in a Lorentzian spacetime (M,g) and knowledge of g|V, can wedetermine (up to diffeomorphism) the spacetime metric g on the domain of causal influence for the set V?
We will show that the answer is yes. In this talk, we will review results and techniques developed in the studyof inverse problems similar to ours. We will also introduce the relativistic Boltzmann equation and comment on
existence of solutions to this PDE given some initial data. We will sketch the key ideas of the proof of our result.
One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the
behaviour of particle collisions captures information about a source-to-solution map for a related linearized
problem. We use this relationship together with an analysis of the behaviour of particle collisions by classicalmicrolocal techniques to determine the set of locations in V where we first receive light particle signals fromcollisions in the unknown domain. From this data we are able to parametrize the unknown region and determine
the metric.
The new results presented in this talk are joint work with Antti Kujanapää, Matti Lassas, and Tony
Liimatainen.

253

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STABILITY OF GEL’FAND’S INVERSE BOUNDARY SPECTRAL PROBLEM VIA
THE UNIQUE CONTINUATION

Dmitri Buragoa, Sergei Ivanovb, Matti Lassasc, Jinpeng Luda
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA,burago@math.psu.edu;b
St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg 191023, Russia,
svivanov@pdmi.ras.ru;cDepartment of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland,matti.lassas@helsinki.fi;dDepartment of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland,
jinpeng.lu@helsinki.fi;

Abstract
Gel’fand’s inverse boundary spectral problem concerns determining the geometry of a compact manifoldwith boundary from the boundary spectral data for the Laplacian. We show that this problem has a stable
solution with quantitative stability estimates in a class of Riemannian manifolds with bounded geometry. Moreprecisely, we show that finitely many Neumann eigenvalues and the boundary values of the corresponding
eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the GromovHausdorff topology. This result is based on an explicit estimate on the stability of the unique continuation for thewave operator from a subset of the manifold boundary.

References
1. D. Burago, S. Ivanov, M. Lassas, J. Lu, Stability of the Gel’fand inverse boundary problem via the uniquecontinuation, arXiv:2012.04435.

254

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

THE FIXED ANGLE SCATTERING PROBLEM WITH A FIRST-ORDERPERTURBATION

Leyter Potenciano-Machado aaDepartment of Mathematics, University of Concepcion, Chileleyter.m.potenciano@gmail.com

Abstract
In this talk, we consider a fixed angle inverse scattering problem associated with the magnetic
Schrödinger operator in dimensions greater or equal than 2. We prove that the magnetic field and the electric
potential are uniquely determined by scattering measurements corresponding to finitely many measurements. We
also show that the number of measurements can be reduced if the coefficients have certain symmetries.
The talk will be based on joint work [1] with Mikko Salo (University of Jyväskylä, Finland) andCristóbal Meroño (Universidad Politécnica de Madrid, Spain).

References
1. C. J. Meroño, L. Potenciano-Machado and M. Salo, The fixed angle scattering problem for the Schrödinger
operator with a first-order perturbation, Ann. Henri Poincaré, 22, 3699-3746 (2021).

255

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SIMULTANEOUS RECOSTRUCTION OF CONDUCTIVITY, BOUNDARY SHAPE
AND CONTACT IMPEDANCES IN EIT

Juan P. Agnellia, Ville Kolehmainenb, Matti Lassasc, Petri Olac and Samuli SiltanencaUniversidad Nacional de Cordoba, FAMAF, Ciudad Universitaria,Cordoba, Argentina, jpagnelli@gmail.com;b
University of Eastern Finland, Dept. of Applied Physics, Kuopio, Finland, ville.kolehmainen@uef.fi;cUniversity of Helsinki, Department of Mathematics and Statistics, 00014 University of Helsinki, Finland,matti.lassas@helsinki.fi, petri.ola@helnki.fi, samuli.siltanen@helsinki.fi;

Abstract
The objective of electrical impedance tomography is to reconstruct the internal conductivity of a physicalbody based on current and voltage measurements at the boundary of the body. In many medical applications theexact shape of the domain boundary and contact impedances are not available. This is problematic as even small
errors in the boundary shape of the computation domain or in the contact impedance values can produce large
artifacts in the reconstructed images, which results in a loss of relevant information. A method is proposed thatsimultaneously reconstructs the conductivity, the contact impedances, and the boundary shape from EIT data.
The approach consists of three steps: first, the unknown contact impedances and an anisotropic conductivity
reproducing the measured EIT data in a model domain are computed. Second, using isothermal coordinates, adeformation is constructed that makes the conductivity isotropic. The final step minimizes the error of true andreconstructed known geometric properties (like the electrode lengths) using conformal deformations.
The feasibility of the method is illustrated with experimental EIT data, with robust and accurate
reconstructions of both conductivity and boundary shape.
This is joint work with Juan Pablo Agnelli, Ville Kolehmainen, Matti Lassas and Samuli Siltanen.

256

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

INVERSE SCATTERING PROBLEMS FOR A BIHARMONIC OPERATOR

Teemu Tyni
University of Toronto, Canada
teemu.tyni@utoronto.ca

Abstract
In this talk we will discuss inverse scattering theory for a first-order perturbation of a biharmonic
operator. The coefficients of the biharmonic operator are vector- and scalar-valued and they can be complex andsingular. We show that the direct scattering problem is well-posed in certain weighted Sobolev spaces undersuitable radiation conditions.
For the inverse problem, the measurement is the scattering amplitude, obtained from the asymptotics ofthe scattering solutions at infinity. We show that in dimensions higher than one it is possible to uniquely recovera combination of the potential functions given the scattering amplitude at all angles at high wave numbers. For
the problem of recovering the potentials from measurements of backscattering, we show the jumps andsingularities of the potentials can be found in dimensions one, two and three. We will also discuss possible
numerical approaches to the problem.
This is a joint work with Valery Serov, Markus Harju and Jaakko Kultima (University of Oulu).

257

MINISYMPOSIUM
M28: Regularization Methods and Applications in Statistics and Econometrics
Organizers:
Pierre Maréchal, University of Toulouse, France, pr.marechal@gmail.com
Anne Vanhems, Toulouse Business School, France, a.vanhems@tbs-education.fr
Ill-posed inverse problems arise in many applications of statistics and econometrics. Typicalexamples are the estimation of a density function (with the deconvolution problem) or of a regressionfunction (with the nonparametric instrumental regression). One main difference with standard cases isthat the operator characterizing the inverse problem is a statistical object that is often unknown andmust be estimated.
The objective of this mini-symposium is to gather experts and young researches to discuss about
recent advances in regularization methods and applications in statistical and econometrics issues.

258

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

TOTAL VARIATION-BASED LAVRENTIEV REGULARISATION OF MONOTONE
PROBLEMS

Fredrik Hildrum and Markus Grasmair
Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, Trondheim,
Norway, {fredrik.hildrum, markus.grasmair}@ntnu.no

Abstract
We study total variation-based (nonlinear) Lavrentiev regularisation of monotone ill-posed problems
A(u) = f .

(1)

Inspired by the efficient taut string-method [1–4] for sparse denoising of piecewise constant spline-signals inwhich A is the identity, we propose algorithms for total-variation denoising with monotone operators A in
Volterra integral equations of the first kind and certain parameter-identification problems.
The new regularisation method is shown to be well-posed for quite general monotone operators andregularisation functionals in Banach spaces, and we also provide some convergence-rate results.

References
1. P. L. Davies and A. Kovac, Local Extemes, Runs, Strings and Multiresolution, The Annals of Statistics, 29(1),1–65, (2001).
2. M. Grasmair, The Equivalence of the Taut String Algorithm and BV-Regularization, Journal of Mathematical
Imaging and Vision, 27(11), 59–66, (2007).
3. L. Condat, A Direct Algorithm for 1-D Total Variation Denoising, IEEE Signal Processing Letters, 20(11),1054–1057, (2000).
4. M. Grasmair and A. Obereder, Generalizations of the Taut String Method, Numerical Functional Analysis andOptimization, 29(3– 4), 346–361 (2008).

259

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

LOCALIZATION OF MOVING SOURCES: UNIQUENESS, STABILITY, AND
BAYESIAN INFERENCE

Mirza Karamehmedovića,*, Faouzi Trikib, and Sára Wangaa
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby,
Denmarkb
Laboratoire Jean Kuntzmann, Université Grenoble Alpes, Saint-Martin-d'Hères, France*
mika@dtu.dk

Abstract
We consider the scalar wave equation in R3 with a moving point source,
,
where
,
,
,
is open, bounded and simply connected, and withvanishing initial conditions. For the direct problem we prove thatis infor any
positive . We also prove uniqueness and stability results for the associated inverse point source problem in thecase the wave field measurement boundary
is real-analytic.
Next, we present and investigate numerically a Bayesian framework for the inference of the source
trajectory and intensity from wave field measurements. The framework employs Gaussian process priors, thepre-conditioned Crank-Nicholson scheme with Markov Chain Monte Carlo sampling, and conditioning onfunctionals to include prior information on the source trajectory.

References
1. G. Hu, Y. Kian, P. Li, Y. Zhao, Inverse moving source problems in electrodynamics, Inverse Problems, 35,
075001 (2019).
2. T. Ohe, Real-time reconstruction of moving point/dipole wave sources from boundary measurements, Inverse
Problems in Science and Engineering, 28 (2020).

260

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON THE DECONVOLUTION OF RANDOM VARIABLES

Pierre Maréchal
Université Paul Sabatier, Toulouse, France
pierre.marechal@math.univ-toulouse.fr;

Abstract
We use mollification to regularize the problem of deconvolution of random variables. This
regularization method provides a unifying framework for comparing the advantages of various filter-like
techniques such as deconvolution kernels, Tikhonov or spectral cutoff. In particular, the mollification approachallows to relax some restrictive assumptions required by deconvolution kernels, and has better stabilizing
properties compared to spectral cutoff or Tikhonov. We show that this approach achieves optimal convergence
rates on the Besov and Sobolev classes. The qualification can be arbitrarily high depending on the choice ofapproximate unit used. We propose an adaptive choice of the regularization parameter using the Lepskii method
and we will illustrate our point by means of numerical simulations.

References
1. T. Hohage, P. Maréchal, L. Simar and A. Vanhems, A mollifier approach to the deconvolution of probability
densities, Working paper, https://dial.uclouvain.be/pr/boreal/object/boreal:259425

261

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

SUPERMIX: SPARSE REGULARIZATION FOR MIXTURES

Y. De Castro, S. Gadat, C. Marteau and C. Maugis-Rabusseau
University of Lyon 1, France, marteau@math.univ-lyon1.fr

Abstract
This contribution investigates the statistical estimation of a discrete mixing measure  0 involved in akernel mixture model. Using some recent advances in  1 -regularization over the space of measures, we
introduce a “data fitting +regularization” convex program for estimating  0 in a grid-less manner, thismethod is referred to as Beurling-LASSO.
Our contribution is two-fold: we derive a lower bound on the bandwidth of our data _tting term
depending only on the support of  0 and its so-called \minimum separation" to ensure quantitative support
localization error bounds; and under a so-called \non-degenerate source condition" we derive a non-asymptotic
support stability property. This latter shows that for sufficiently large sample size n , our estimator has exactly
as many weighted Dirac masses as the target  0 , converging in amplitude and localization towards the true
ones. The statistical performances of this estimator are investigated designing a so-called “dual certificate”,
which will be appropriate to our setting. Some classical situations, as e.g. Gaussian or ordinary smooth mixtures(e.g. Laplace distributions), will be discussed.
We stress in particular that our method is completely adaptive w.r.t. the number of components
involved in the mixture.

262

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

ON HAMILTON-JACOBI PDES AND IMAGE DENOISING MODELS WITH
CERTAIN NON-ADDITIVE NOISE

Elena Resmerita
Institute of Mathematics, University of Klagenfurt, Austria
elena.resmerita@aau.at

Abstract
We consider image denoising problems formulated as variational problems. It is known that HamiltonJacobi PDEs govern the solution of such optimization problems when the noise model is additive. In this work,
we address certain non-additive noise models and show that they are also related to Hamilton-Jacobi PDEs.
These findings allow us to establish new connections between additive and non-additive noise imaging models.
With these connections, some non-convex models for non-additive noise can be solved by applying convex
optimization algorithms to the equivalent convex models for additive noise.
Several numerical results are provided for denoising problems with Poisson noise or multiplicative
noise.
This is joint work with Jerome Darbon and Tingwei Meng.

263

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

REGULARISATION, OPTIMISATION, SUBREGULARITY

Tuomo Valkonena,baEscuela Politenica Nacional, Quito, Ecuador, tuomo.valkonen@iki.fi;bDepartment of Mathematics and Statistics, University of Helsinki, Finland

Abstract
Regularisation theory in Banach spaces, and non-norm-squared regularisation even in finite dimensions,generally relies upon Bregman divergences to replace norm convergence. This is comparable to the extension of
first-order optimisation methods to Banach spaces. Bregman divergences can, however, be somewhat suboptimalin terms of descriptiveness. Using the concept of (strong) metric subregularity, previously used to prove the fast
local convergence of optimisation methods, we show norm convergence in Banach spaces and for non-normsquared regularisation. For problems such as total variation regularised image reconstruction, the metricsubregularity reduces to a geometric condition on the ground truth: flat areas in the ground truth have tocompensate for the fidelity term not having second-order growth within the kernel of the forward operator. Our
approach to proving such regularisation results is based on optimisation formulations of inverse problems.
As a side result of the regularisation theory that we develop, we provide regularisation complexity
results for optimization methods: how many steps Nδ of the algorithm do we have to take for the approximate
solutions to converge as the corruption level δ ↘ 0?

References
1. D. Burago, S. Ivanov, M. Lassas, J. Lu, Stability of the Gel’fand inverse boundary problem via T. Valkonen,
Regularisation, optimization, subregularity, Inverse Problems, 37, 045010(2021).

264

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

A MOLLIFIER APPROACH TO THE NONPARAMETRIC INSTRUMENTAL
REGRESSION PROBLEM

Anne Vanhems
TBS Education, Toulouse, France
a.vanhems@tbs-education.fr;

Abstract
We propose a mollification method for the problem of instrumental regression. We show that ourestimator is consistent, and compare its performances to that of the classical regularization methods. A finite
sample study enables us to demonstrate the efficiency of mollification compared to other estimation methods.

References
1. Xavier Bonnefond and Pierre Marechal. A variational approach to the inversion of some compact operators.
Pacific journal of optimization, 5(1) (2009).
2. Marine Carrasco, Jean-Pierre Florens, and Eric Renault. Linear inverse problems in structural econometrics
estimation based on spectral decomposition and regularization. Handbook of econometrics, 6, (2007).

265

Poster Session

266

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

STABILITY FOR A SPECIAL CLASS OF ANISOTROPIC CONDUCTIVITIES VIAAN AD-HOC MISFIT FUNCTIONAL
Sonia Foschiattia , Romina Gaburrob and Eva Sincichaa Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, via Valerio 12/1, 34127, Trieste,
Italy, sonia.foschiatti@phd.units.it;b Department of Mathematics and Statistics, University of Limerick, Limerick, V94 T9PX, Ireland,romina.gaburro@ul.ie;

Abstract
In recent years there has been an increasing interest in the Calderòn problem with anisotropic
conductivity. In particular, the geometric nature of the problem has been extensively studied to determine when
uniqueness and stability occur. In our latest work, we have followed a different line of research, inspired by theideas introduced by Alessandrini and Vessella in [1], where they proved Lipschitz stability for piecewise
constant isotropic conductivities. We have managed to extend their argument to a special class of anisotropic
conductivities.
In this poster we are going to review the stability estimate in terms of a novel misfit functional and thenderive as a corollary a Lipschitz stability estimate in terms of the classical local Dirichlet-to-Neumann map.

References
1. G. Alessandrini, S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. Appl. Math. 35,
207-241 (2005).

267

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

JACOBIAN OF SOLUTIONS TO THE CONDUCTIVITY EQUATION IN LIMITED
VIEW
Mikko Saloa and Hjørdis Amanda Schlüterb
a Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, 40014, Finland,
mikko.j.salo@jyu.fi;b Department of Applied Mathematics and Computer Science, Technical University of Denmark, RichardPetersens Plads, Kgs. Lyngby, 2800, Denmark, hjsc@dtu.dk

Abstract
We address the Jacobian of two solutions to the conductivity equation in a limited view setting for adomain in two-dimensional Euclidean space. The limited view setting is characterized by a part of the boundarythat we can control with a Dirichlet boundary condition and a part of the boundary, where the solution is zero.
Our work is motivated by the Radó-Kneser-Choquet theorem which states that under certain conditions on theboundary functions the Jacobian of the solutions is non-vanishing. This result has been generalized from the caseof a constant isotropic conductivity to anisotropic conductivities. The key novelty of the presented work is thegeneralization of this theorem from the full view to the limited view setting for anisotropic conductivities. We
show under which condition on the boundary function the corresponding solution has no critical points.
Moreover, how this result can be extended so that the Jacobian of two solutions is non-vanishing. This is related
to the number of arcs on the boundary along which the boundary functions are non-increasing and nondecreasing.
This type of result is used in inverse problems for reconstructing the conductivity from interior power
densities. Here the non-vanishing Jacobian condition ensures invertability of the matrix composed of power
densities.
We illustrate how this result can be used for reconstruction of an isotropic conductivity from power
densities by a numerical example.

268

The 10th International Conference”Inverse Problems: Modeling and Simulation”(IPMS-2022)
Malta, May 22 - 28, 2022

OPTIMAL EXPERIMENTAL DESIGN FOR SOUND SOURCE LOCALIZATION
Phuoc Truong Huynha
a Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria,
phuoc.huynh@aau.at

Abstract
Acoustic source localization has a variety of application areas. Especially, methods using the inverse
scheme with processed measurement data captured by microphone arrays have been commonly used over thelast decade.
In this poster, we propose a method for optimal experimental design for the acoustic source
identification problem. Optimizing the design of experiments is important so that the uncertainty in the estimatedparameters is minimized. The source that needs to be identified is sparse and infinite-dimensional, which is
challenging from both mathematical and computational points of view. Our approach relies on an iteratively
reweighted least-squares scheme.
This is a joint work with Barbara Kaltenbacher, Daniel Walter and Konstantin Pieper.

269