A numerical scheme for severely ill-posed
nonlinear inverse problems with a regularized
Moore-Penrose pseudoinverse
Manabu Machida
Department of Informatics, Faculty of Engineering, KindaiUniversityInverse problems appearing in optical tomography are nonlinear and severely
ill-posed. Among non-iterative schemes, the Born and Rytov approximations
are well known. These approximations, however, require linearization of thenonlinear inverse problem. Then the linearized inverse problem can be solved
using the Moore-Penrose pseudoinverse with a suitable regularization. In thistalk, I will take the nonlinearity of the inverse problem into account by considering the inversion of the Born and Rytov series. Since the Rytov approximation
is more practical and has been used for experiment and clinical research, inparticular I will explore nonlinear Rytov approximations by constructing theinverse Rytov series.

Inferring properties of the human brain fromclinical and experimental data
Stephen John Payne
Institute of Applied Mechanics and Department of Mathematics,
National Taiwan University

The human brain is one of the most complicated systems known to humankind. With around 85 billion neurons and nearly 100,000 miles of bloodvessels, it is a highly complex, self-organised, active system. However, measuring brain structure and function is surprisingly difficult. Despite major advancesin imaging technologies, our understanding of both the structure and the function of the brain remains surprisingly poor. Mathematical models thus play
an important role in interpreting clinical and experimental data. I will present
work in my group that has led to the development of multiple scale methods
to simulate cerebral blood flow, oxygen transport, and water movement. These
have now been used to simulate these processes in whole-brain models for the
first time within a mathematically rigorous, yet computationally inexpensive
framework.
A key difficulty with these models, however, remains the issue of parameter
estimation. I will present our preliminary work on parameter estimation and theuse of optimisation techniques, together with data from a potential new imaging
modality. However, much work remains to be done to obtain robust estimatesof model parameters across different length scales from both experimental data
(both human and animal) and clinical data. Potential future avenues for this will
be discussed with a view to drawing on the expertise of others at the workshop.

Control of discrete-time stochastic systems andapplication to remote automated driving
Yohei Hosoe
Department of Electrical Engineering,
Graduate School of Engineering, Kyoto University

This talk is mainly about control theory for discrete-time systems whosestate transition is determined randomly. In the linear case, this kind of stochastic systems can be described by state equations with coeﬃcient matrices given
by discrete-time matrix-valued stochastic processes. In this talk, recent results
on second-moment stability of the stochastic systems and associated Lyapunov
inequalities are ﬁrst introduced, without restricting the class of the associatedstochastic processes. Then, some theoretical extensions are further discussed.
As one of the possible practical applications of such stochastic control theory,
results obtained in university-industry collaborative research on remote automated driving are also introduced.

Globally injective and bijective neural operators
Takashi FURUYA ( Shimane University)∗
Recently, there has been great interest in neural operators [2], which learn operators between infinite-dimensional function spaces. Given their discretization-invarianceproperty, neural operators have been remarkably successful, especially in learning solution operators for partial differential equations (PDEs). In this talk, we show that
injective neural operators are universal approximators of continuous operators. This
holds substantial relevance to inverse problems in PDEs, as injective neural operators
can serve invertible surrogate models for solution operators, which, in general, is notinjective due to ill-posnedss of inverse problems. This talk bases on [1], which is ajoint work with Michael Puthawala (South Dakota State University), Matti Lassas(University of Helsinki), and Maarten V. de Hoop (Rice University).

References[1] Takashi Furuya, Michael Puthawala, Matti Lassas, and Maarten V de Hoop. Globally
injective and bijective neural operators. arXiv preprint arXiv:2306.03982, 2023.
[2] Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps
between function spaces. arXiv preprint arXiv:2108.08481, 2021.

Keywords: Deep Learning, Operator Learning, Functional Analysis, Partial Differential Equations,
Injectivity, Universal approximation.
∗e-mail: takashi.furuya0101@gmail.com

Stability for nonlinear inverse problems with low
dimensional priors
Matteo Santacesaria
Department of Mathematics, University of Genoa

In this talk I will review some stability properties of a large class of inverse
problems in infinite dimensional (Banach) spaces, especially nonlinear and illposed (e.g. electrical impedance tomography). The main focus will be on how
to improve stability by imposing low dimensional priors on the unknown, in a
deterministic setting, using functional analytic techniques. The low dimensional
priors considered will be mainly finite dimensional linear subspaces and finite
dimensional manifolds. These results can be combined with manifold learningtechniques (in particular based on generative models) to learn a low dimensionalparameterization of the unknown, which yields again stability under suitableconditions. This is based on a series of works done in collaboration with G.S.
Alberti (University of Genoa), A. Arroyo (Complutense Madrid), J. Hertrich(TU Berlin), S. Sciutto (University of Genoa).

Inverse problems of nonlinear wave equations
Xi ChenShanghai Centre for Mathematical Sciences, Fudan University

We shall give a survey of some nonlinear wave equations and inverse problems
arising in mathematical physics. In particular, we shall present recent progresson stable recovery of coefficients of nonlinear wave equations.

Bayesian inversion in a trans-dimensional framework forsubsurface stratification
Michael Conrad Koch
Graduate School of Agriculture, Kyoto University, Kyoto, Japan

Abstract

Subsurface stratification from Cone Penetration Test (CPT) data to delineate different
soil layers and their layer boundaries is an important task in soil investigation exercisesin agricultural, geotechnical and geological engineering. The stratification parameters
are: the number of soil layers, the layer interface depths and the soil spatial random
field parameters. Seeing that the number of layers is unknown apriori and that theother two parameters are dependent on the number of layers, the problem assumes atrans-dimensional characteristic where the dimension of the parameter space can change.
Using a Bayesian formulation, a simple proposal density is designed that satisfies thereversibility criteria in the Reversible Jump Markov Chain Monte Carlo (RJMCMC)
framework. The proposal density enables jumping between parameter spaces of different dimensions and forms part of the first sampling block in a three-block MCMC
algorithm. The other two blocks involve sampling of the spatial random field parameters. RJMCMC algorithms typically display poor acceptance rates necessitating thedevelopment of efficient samplers. The particular blocking strategy described allows forthe development of a formulation that primarily involves computationally inexpensivetasks such as sampling from truncated normal and Inv-Gamma distributions and evaluation of general normal densities. The algorithm is validated using synthetic and realworld CPT data. A standard normality check of the decorrelated residuals is seen as a
useful measure to test algorithm performance.

Identification of Conductivity in Elliptic
equations using Deep Neural Networks
Zhi ZhouDepartment of Applied Mathematics, The hong Kong Polytechnic
UniversityThe focus of this talk is on the numerical methods used to identify the conductivity in an elliptic equation. Commonly, a regularized formulation consistsof a data fidelity and a regularizer is employed, and then it is discretized using
finite difference method, finite element methods or deep neural networks. Onekey issue is to establish a priori error estimates for the recovered conductivity
distribution. In this talk, we discuss our recent findings on using deep neural networks for this class of problems, by effectively utilizing relevant stability
estimates.

The field concentration problem in nano-optics
Sanghyeon Yu
Department of Mathematics, Korea University

Composite materials shows the high field concentration when the inclusions
get close to touching. This phenomenon has many practical applications inimaging, spectroscopy, and meta-materials. In this talk, we discuss a new
way of tackling the field concentration problem via the spectral analysis of the
Neumann-Poincare operator. We shall also discuss a related inverse problem.

Modeling of slip on a fault during an earthquake:point-source approximation
Shiro Hirano
Ritsumeikan University (s-hrn@fc.ritsumei.ac.jp)

Some parts of the Earth’s crust dynamically rupture and slip during earthquakes, and their deformation is often represented by the equation of motion ofan elastic domain Ω(∋ x):((()))ρ ü =∇ · λ (∇ · u) I + µ ∇u + ∇u)T(1)where u = u(x, t) is the displacement vector, and ρ = ρ(x), λ = λ(x), andµ = µ(x) are the density and Lamé parameters in the heterogeneous Earth,respectively. While the domain Ω is a continuum, the velocity during the coseismic slip must be discontinuous across the ruptured region Γ(⊂ Ω), calleda fault. Seismologists focus on the gap of displacement and velocity across Γ.
Here, we assume that Γ is planar and lying along x1 -x2 -plane, and the gap isparallel to x1 -axis, for simplicity. Then, the slip at s(∈ Γ) is deﬁned as:D(s, t) := lim [u1 (s + εν, t) − u1 (s − εν, t)] ,ε↓0(2)where ν = (0, 0, 1)T is a normal vector to Γ.
The forward and inverse problems to model slip D and slip velocity Ḋ area major topic in seismology. When we observe ground velocity u̇ at suﬃciently
far ﬁeld, the spatial extent of Γ would be negligible, and u̇ is syntheticallyrepresented as follows:∫ tu̇(x, t) =Ḋ(τ ) G(x, t − τ ; s) dτ,(3)0

where G is a propagator obtained synthetically or empirically. Some empiricallaws are known for the moment and spectra of Ḋ in this point-source approximation. The author will discuss their main points and a stochastic model torepresent the empirical laws.

References[1] Hirano, S. (2022). Source time functions of earthquakes based on a stochastic diﬀerential equation, Scientiﬁc Reports, 12:3936, https://doi.org/10.1038/s41598-022-07873-2

Scattering analysis of guided wave beam bydefects in a plate with finite width
Taizo Maruyama
Department of Civil and Environmental Engineering, Tokyo
Institute of Technology

Guided wave scattering arises in nondestructive testing (NDT) using ultrasonic guided waves. The guided waves display dispersion and multimodal
characteristics, making it crucial to employ dispersion analysis to understandthem [1]. Additionally, scattering analysis necessary to comprehend the received
waves in NDT. Typically, numerical methods are used to conduct dispersion andscattering analyses. This study proposes a hybrid semi-analytical finite elementboundary element method (SAFE-BEM) for this purpose.
The dispersion curves for a three-dimensional plate with finite width arecomplex, making it challenging to directly employ them for the analysis of incident and received waves in experimental measurement. In conventional guided
wave measurement, the incident and received waves are considered using twodimensional dispersion curves for an infinite-width plate [2]. To understand theincident and received waves in terms of the two-dimensional dispersion properties, we examine the appropriate incident wave and scattered far-fields innumerical simulations. For this purpose, the Gaussian beam of plate wave [3]is considered as an incident wave.

References[1] Rose, J. L.: Ultrasonic Guided Waves in Solid Media, Cambridge University
Press, 2014.
[2] Chen, Q., Xu, K., and Ta, D.: High-resolution Lamb waves dispersion
curves estimation and elastic property inversion, Ultrasonics 115, 106427,2021.
[3] Maruyama, T. and Kanda, K.: Derivation of an in-plane amplitude equation and its paraxial approximation for elastic guided waves in plates, International Journal of Solids and Structures 281, 112420, 2023.

Dual-grid parameter choice method for total
variation regularized image deblurring
Markus Juvonen
Department of Mathematics and Statistics, University of Helsinki

I will talk about a new parameter choice method for total variation (TV)deblurring of images which is based on a dual-grid computation of the solution.
Instead of a single grid we have two grids with different discretisation. The
first grid is the same were the measurement is given. The origin of the secondgrid is shifted half a pixel width both horizontally and vertically. Note that theunderlying true image is the same for both grids. Assume that the pixel size
is much smaller than a typical constant valued area in an image. The premiseof the study is that when solving the TV regularised noisy deblurring problem
with a large enough parameter the solutions on both grids will converge to thesame image. The proposed algorithm looks for the smallest parameter withwhich convergence can be numerically detected. The method has been tested
on both simulated and real image data. Computational experiments suggestthat an optimal parameter can be chosen by monitoring the relative differenceof the TV seminorms of the dual-grid solutions while changing the regularisation
parameter size. This talk is based on collaboration with Yiqiu Dong (DTU),
Samuli Siltanen (University of Helsinki), Matti Lassas (University of Helsinki)
and Ilmari Pohjola (University of Helsinki).

Electrical impedance tomography and virtual
X-raysSamuli, Siltanen
Industrial Mathematics, University of Helsinki

A connection between Electrical Impedance Tomography (EIT) and X-ray
tomography was found in [Greenleaf et al. 2018] using microlocal analysis.
Fourier transform applied to the spectral parameter of Complex Geometric Optics solutions produces virtual X-ray projections, enabling a novel filtered backprojection type nonlinear reconstruction algorithm for EIT. This approach iscalled Virtual Hybrid Edge Detection. It is remarkable how this new approachdecomposes the EIT image reconstruction process in several steps, where all
ill-posedness is confined in two linear steps. Therefore, we can separate thenonlinearity and ill-posedness of the fundamental EIT problem.

Approximate peak time and its application totime-domain fluorescence diffuse optical
tomography
Shuli Chen
Department of Mathematics, Hokkaido University

This talk will present an inverse problem for the fluorescence diffuse optical
tomography (FDOT) identifying multiple point targets. The targets are thefluorescent agents. The FDOT process is modeled by two diffusion equations
coupled with the source term. Our measured data are the time-domain data,measured at some detection points as the temporal response of the fluorescenceintensity to an instantaneous injection of the excitation light from a source
point. The peak time, clearly observed in the space-time data, has been used as
an index to detect the targets. We introduce an approximate peak time basedon an asymptotic analysis, which agrees very well with the peak time obtainedby the numerical calculation using typical optical parameters. Then, using approximate peak time and combining it with the bisection method, we propose amathematically rigorous inversion method for the FDOT. The proposed method
is efficient and robust and accurate for identifying target locations embedded indeep depth. This is a joint work with Junyong Eom, Gen Nakamura and Goro
Nishimura.

A Mathematical Theory of ComputationalResolution Limit
Hai Zhang
Department of Mathematics, The Hong Kong University of Scienceand Technology

It is well-known that the resolution of optical imaging system is fundamentally limited by the optical wavelength. Based on this, Rayleigh proposed
the Rayleigh criterion on the minimum resolvable distance between two point
sources, the so called Rayleigh limit. Although widely used in the practice, thislimit is not so useful for images that are subject to elaborated data processing.
To remedy this, we develop a theory of computational resolution limit to characterize the fundamental resolution limit from the approximation theory pointof view. The theory can be used to explain the phase transition phenomenon
in the reconstruction problem. New efficient super-resolution algorithm is also
developed following the theory.

On numerical instability of a singular integral
equation in x-ray computerized tomography withpartial measurement
Hiroshi Fujiwara (Kyoto Univ.), Kamran Sadiq (RICAM), andAlexandru Tamasan (University of Central Florida)

We consider x-ray tomography with limited data measured only on an arcof the boundary. A Cauchy-type boundary integral formula has been proposedby Bukhgeim (1995), and the jump relation applied to it derives a Cauchy-typesingular integral equation. It is well known that its iversion is unstable in the L2
sense even if it exists. This talk presents an quantitative estimate of instability
under some discretization, and it shows that the instability is not severe. Somenumerical examples are exhibited to support our analysis and to show feasibilityof the proposed algorithms. We note that the proposed reconstruction procedureis neither statistical nor iterative.

References[1] . L. Bukhgeim, Inversion formulas in inverse problems, in Linear operatorsand ill-posed problems, Plenum, New York (1995) 323–378.
[2] . Fujiwara, K. Sadiq, and A. Tamasan, Numerical Reconstruction of Radiative Sources from Partial Boundary Measurement, SIAM J. Imaging Sci.
16 (2023) 948–968.