construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography we explain how to build indistinguishable isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two dimensional electrical impedance tomography the theoretical approach based on solving a fixed point problem is constructive and allows the implementation of an algorithm for approximating the invisible perturbations the functionality of the method is demonstrated via numerical examples

compensation of modelling errors due to unknown domain boundary in electrical impedance tomography electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors this instability is especially severe when absolute imaging is considered with clinical measurements accurate knowledge about the body shape is usually not available and therefore an approximate model domain has to be used in the computational model it has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect in this paper we adapt the so called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape this approach has previously been shown to be applicable to a variety of modeling errors such as coarse discretization in the numerical approximation of the forward model and domain truncation we evaluate the approach with a simulated example of thorax imaging and also with experimental data from a laboratory setting with absolute imaging considered in both cases we show that the related modeling errors can be efficiently compensated for by the approximation error approach we also show that recovery from simultaneous discretization related errors is feasible allowing the use of computationally efficient reduced order models

weighted regularization in electrical impedance tomography with applications to acute cerebral stroke we apply electrical impedance tomography to detect and localize brain impedance changes associated with stroke forward solutions are computed using the finite element method in two dimensions we assume that baseline conductivity values are known for the major head tissues and focus on changes in the brain compartment only we use singular value decomposition svd to show that different impedance measurement patterns which are theoretically equivalent by the reciprocity theorem have different sensitivities to the brain compartment in the presence of measurement noise the inverse problem is solved in part by standard means using iterated svd and regularizing by truncation to improve regularization we introduce a weighting scheme which normalizes the sensitivity matrix for voxels at different depths this increases the number of linearly independent components which contribute to the solution and forces the different measurement patterns to have similar sensitivity when applied to stroke this weighted regularization improves image quality overall

the factorization method applied to the complete electrode model of impedance tomography the factorization method is a tool for recovering inclusions inside a body when the neumann to dirichlet operator which maps applied currents to measured voltages is known in practice this information is never at hand due to the discreteness and physical properties of the measurement devices the complete electrode model of impedance tomography includes these physical characteristics but leads to a finite dimensional data set called the resistivity matrix the main result of this work is an approximation link relating the resistivity matrix to the neumann to dirichlet operator in the l 2 operator norm this result allows us to extend the factorization method to the framework of real life electrode measurements using a regularized series criterion which is easy to implement in practice the truncation index of the sequence criterion which represents the stopping index of the regularization scheme can be computed solely from the measured perturbed and finite dimensional data the functionality of

stochastic algorithms for inverse problems involving pdes and many measurements inverse problems involving systems of partial differential equations pdes can be very expensive to solve numerically this is so especially when many experiments involving different combinations of sources and receivers are employed in order to obtain reconstructions of acceptable quality the mere evaluation of a misfit function the distance between predicted and observed data often requires hundreds and thousands of pde solves this article develops and assesses dimensionality reduction methods both stochastic and deterministic to reduce this computational burden we assume that all experiments share the same set of receivers and concentrate on methods for reducing the number of combinations of experiments called simultaneous sources that are used at each stabilized gauss newton iteration algorithms for controlling the number of such combined sources are proposed and justified evaluating the misfit approximately except for the final verification for terminating the process always involves

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

reconstruction of biologic tissue conductivity from noisy magnetic field by integral equation method abstract magnetic resonance electrical impedance tomography mreit is a new technique to recover the conductivity of biologic tissue from the induced magnetic flux density this paper proposes an inversion scheme for recovering the conductivity from one component of the magnetic field based on the nonlinear integral equation method to apply magnetic fields corresponding to two incoherent injected currents an alternative iteration scheme is proposed to update the conductivity for each magnetic field the regularizing technique on the finite dimensional space is applied to solve an ill posed linear system compared with the well developed harmonic b z method the advantage of this inversion scheme is its stability since no differential operation is required on the noisy magnetic field numerical implementations are given to show the convergence of the iteration and its validity for noisy input data

effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary the linearized problem was studied by calderon who showed that if the conductivity differs little from a constant then it can be reconstructed approximately in his work calderon assumed that arbitrary current densities can be applied anywhere on the body s surface and the resulting voltages perfectly measured in practice however imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision in this paper linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements the effects of finite precision and electrode number on a two dimensional version of calderon s method are studied it is shown that as the measurement errors go to zero and the electrodes proliferate in a space filling manner the finite electro

inversion of 3d electromagnetic data in frequency and time domain using an inexact all at once approach we present a general formulation for inverting frequencyor time domain electromagnetic data using an all at once approach in this methodology the forward modeling equations are incorporated as constraints and thus we need to solve a constrained optimization problem where the parameters are the electromagnetic fields the conductivity model and a set of lagrange multipliers this leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought nevertheless experience shows that the constrained problem can be solved faster than the unconstrained one the primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly in this paper we outline the all atonce approach and apply it to electromagnetic problems in both frequency and time domains this is facilitated by a unified representation for forward modeling for these two types of data the optimization problem is solved by finding a stationary point of the lagrangian numerically this leads to a nonlinear system that is solved iteratively using a gauss newton strategy at each iteration a large indefinite matrix is inverted and we discuss how this can be accomplished as a test we invert frequency domain synthetic data from a grounded electrode system that emulates a field csamt survey for the time domain we invert borehole data obtained from a current loop on the surface

on effective methods for implicit piecewise smooth surface recovery this paper considers the problem of reconstructing a piecewise smooth model function from given measured data the data are compared to a field which is given as a possibly nonlinear function of the model a regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities regularization operators related to total variation tv are therefore employed two popular methods are modified tv and huber s function both contain a parameter which must be selected the huber variant provides a more natural approach for selecting its parameter and we use this to propose a scheme for both methods our selected parameter depends both on the resolution and on the model average roughness thus it is determined adaptively its variation from one iteration to the next yields additional information about the progress of the regularization process the modified tv operator has a smoother generating function nonetheless we obtain a huber variant with comparable and occasionally better performance for large problems e g high resolution the resulting reconstruction algorithms can be tediously slow we propose two mechanisms to improve efficiency the first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution the second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

reconstructing discontinuities using complex geometrical optics solutions in this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the laplacian as the leading order term we apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements computational results demonstrate the versatility of these solutions to determine electrical inclusions

numerical solution method for the dbar equation in the plane a fast method for solving equations of the form v tv is presented where v and t are complex valued functions of two real variables the multigrid method of vainikko int soc anal appl comput 5 2000 is adapted to the problem with a fft implementation convergence with rate o h is proved for the method applied to equations of the form above one grid and two grid versions of the method are implemented and their effectiveness is demonstrated on an application arising in electrical impedance tomography eit

finite element implementation of maxwell s equations for image reconstruction in electrical impedance tomography traditionally image reconstruction in electrical impedance tomography eit has been based on laplace s equation however at high frequencies the coupling between electric and magnetic fields requires solution of the full maxwell equations in this paper a formulation is presented in terms of the maxwell equations expressed in scalar and vector potentials the approach leads to boundary conditions that naturally align with the quantities measured by eit instrumentation a two dimensional implementation for image reconstruction from eit data is realized the effect of frequency on the field distribution is illustrated using the high frequency model and is compared with laplace solutions numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation the results show that scalar vector potential reconstruction produces images which are essentially indistinguishable from a laplace algorithm for frequencies below 1 mhz but superior at frequencies reaching 10 mhz

numerical solution of the cauchy problem for the stationary schrodinger equation using faddeev s green function numerical solution of the cauchy problem for the stationary schrodinger equation in a bounded two dimensional domain is discussed the solution algorithm is based on the properties of faddeev s green function numerical examples with computer simulated data are presented including an application to the inverse potential problem of electrocardiography

on the solvability of the discrete conductivity and schroźdinger inverse problems we study the uniqueness question for two inverse problems on graphs both problems consist in finding possibly complex edge or nodal based quantities from boundary measurements of solutions to the dirichlet problem associated with a weighted graph laplacian plus a diagonal perturbation the weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete schro dinger potential we use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity or schro dinger potential then the linearized problem is solvable for almost all conductivities or schro dinger potentials in a suitable set we show that the conductivities or schro dinger potentials in a certain set are determined uniquely by boundary data except on a zero measure set this criterion for solvability is used in a statistical study of graphs where the conductivity or schro dinger inverse problem is solvable

uniqueness results for inverse robin problems with bounded coefficient in this paper we address the uniqueness issue in the classical robin inverse problem on a lipschitz domain omega subset rr n with l infty robin coefficient l 2 neumann data and isotropic conductivity of class w 1 r omega r n we show that uniqueness of the robin coefficient on a subpart of the boundary given cauchy data on the complementary part does hold in dimension n 2 but needs not hold in higher dimension we also raise on open issue on harmonic gradients which is of interest in this context

krylov subspace iterative techniques on the detection of brain activity with electrical impedance tomography in this paper we review some numerical techniques based on the linear krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems exploring their computational advantages in solving large scale systems of equations we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain if the conductivity of the head tissues is assumed to be real the preconditioned conjugate gradients pcgs algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance the performance and the regularizing properties of the pcg iteration for solving ill conditioned systems of equations pcgns is then explored and a suitable preconditioning matrix is suggested in order to enhance its convergence rate for image reconstruction the nonlinear inverse problem is considered based on the gauss newton method for solving nonlinear problems we have developed two algorithms that implement the pcgn iteration to calculate the linear step solution using an anatomically detailed model of the human head and a specific scalp electrode arrangement images of a simulated impedance change inside brain s white matter have been reconstructed

application of stochastic galerkin fem to the complete electrode model of electrical impedance tomography abstract the aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage the most accurate forward model for impedance tomography is the complete electrode model which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces if the reconstruction task of impedance tomography is recast as a bayesian inference problem it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables respectively in this work we apply a stochastic galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with monte carlo simulations

an a posteriori driven adaptive mixed high order method with application to electrostatics in this work we propose an adaptive version of the recently introduced mixed high order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism the latter include in particular the numerical modeling of comb drive and mems devices mesh adaptation is driven by newly derived residual based error estimators the resulting method has several advantageous features it supports fairly general meshes it enables arbitrary approximation orders and has a moderate computational cost thanks to hybridization and static condensation the a posteriori driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions allowing to fully exploit the high order of approximation

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

crack reconstruction using a level set strategy we present a level set based technique to recover key characteristics of a defect or crack e g location length and shape in a two dimensional material from boundary electrical measurements the key feature of this work is to extend the usual level set technique for modeling volumetric objects to very thin objects two level set functions are employed the first one models the location and form of the crack and the second one models its length and connectivity an efficient gradient based method is derived in order to define evolution laws for these two level set functions which minimize the least squares data misfit numerical experiments show the utility of this method even in the presence of a significant noise level in the measurements a finite element method is used to simulate the electric field behavior in the presence of very thin objects

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

spectral approach to the scattering map for the semi classical defocusing davey stewartson ii equation abstract the inverse scattering approach for the defocusing davey stewartson ii equation is given by a system of d bar equations we present a numerical approach to semi classical d bar problems for real analytic rapidly decreasing potentials we treat the d bar problem as a complex linear second order integral equation which is solved with discrete fourier transforms complemented by a regularization of the singular parts by explicit analytic computation the resulting algebraic equation is solved either by fixed point iterations or gmres several examples for small values of the semi classical parameter in the system are discussed

fast segmentation from blurred data in 3d fluorescence microscopy we develop a fast algorithm for segmenting 3d images from linear measurements based on the potts model or piecewise constant mumford shah model to that end we first derive suitable space discretizations of the 3d potts model which are capable of dealing with 3d images defined on non cubic grids our discretization allows us to utilize a specific splitting approach which results in decoupled subproblems of moderate size the crucial point in the 3d setup is that the number of independent subproblems is so large that we can reasonably exploit the parallel processing capabilities of the graphics processing units gpus our gpu implementation is up to 18 times faster than the sequential cpu version this allows to process even large volumes in acceptable runtimes as a further contribution we extend the algorithm in order to deal with non negativity constraints we demonstrate the efficiency of our method for combined image deconvolution and segmentation on simulated data and on real 3d wide field fluorescence microscopy data

statistically stable ultrasonic imaging in random media analysis of array data from acoustic scattering in a random medium with a small number of isolated targets is performed in order to image and thereby localize the spatial position of each target because the host medium has random fluctuations in wave speed the background medium is itself a source of scattered energy it is assumed however that the targets are sufficiently larger and or more reflective than the background fluctuations so that a clear distinction can be made between targets and background scatterers in numerical simulations nonreflective boundary conditions are used so as to isolate the effects of the host randomness from those of the spatial boundaries which can then be treated in a separate analysis it is shown that the key to successful imaging is finding statistically stable functionals of the data whose extreme values provide estimates of scatterer locations the best ones are related to the eigenfunctions and eigenvalues of the array response matrix just as one might expect from prior work on array data processing in complex scattering media having homogeneous backgrounds the specific imaging functionals studied include matched field processing and linear subspace methods such as music mutiple signal classification but statistical stability is not characteristic of the frequency domain which is often the province of these methods by transforming back into the time domain after first diagonalizing the array data in the frequency domain one can take advantage of both the time domain stability and the frequency domain orthogonality of the relevant eigenfunctions

d bar method for electrical impedance tomography with discontinuous conductivities the effects of truncating the approximate scattering transform in the d bar reconstruction method for two dimensional electrical impedance tomography are studied the method is based on the uniqueness proof of nachman ann of math 2 143 1996 pp 71 96 that applies to twice differentiable conductivities however the reconstruction algorithm has been successfully applied to experimental data which can be characterized as piecewise smooth conductivities the truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity thus the truncation can be interpreted as regularization of the d bar method numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the d bar method further a new connection between calderon s linearization method and the d bar method is established and the two methods are compared numerically and analytically

a kaczmarz version of the reginn landweber iteration for ill posed problems in banach spaces in this work we present and analyze a kaczmarz version of the iterative regularization scheme reginn landweber for nonlinear ill posed problems in banach spaces q jin inverse problems 28 2012 065002 kaczmarz methods are designed for problems which split into smaller subproblems which are then processed cyclically during each iteration step under standard assumptions on the banach space and on the nonlinearity we prove stability and norm convergence as the noise level tends to zero further we test our scheme on the inverse problem of two dimensional electric impedance tomography not only to illustrate our theoretical findings but also to study the influence of different banach spaces on the reconstructed conductivities

separable laplace equation magic toeplitz matrix and generalized ohm s law we say that a laplace equation on an inhomogeneous medium is separable if the medium can be represented as the product of the argument functions it is shown that the conductivity equation on a 2 d disk is reduced to a couple of ordinary differential equations if the conductivity of the medium is the product of the angle and radius functions several important cases are discussed such as circular and a noncircular blob like medium conductivity we derive a closed form of the neumann to dirichlet map that gives rise to a new family of symmetric matrices we call it the magic toeplitz matrix a symmetric toeplitz matrix with the rows and columns summing to the same value the generalized ohm s law on the disk is derived which relates the vector of the current density to the vector of voltages through the resistance matrix we demonstrate how a separable laplace pde can be applied to electrical impedance tomography and breast cancer detection

a real time algorithm for the location search of discontinuous conductivities with one measurement we consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current voltage observation we develop a real time algorithm for determining the location of the anomaly this new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage combined with the size estimation result this algorithm gives a good initial guess for newton type schemes we give the rigorous proof for the location search algorithm both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast stable and efficient 2001 john wiley sons inc

efficient electric resistivity inversion using adjoint state of mixed finite element method for poisson s equation we propose an electric resistivity inversion method that is similar to the reverse time migration technique applied to seismic data for calculating model responses and inversion we use the mixed finite element method with the standard p 1 p 0 pair for triangular decompositions which makes it possible to compute both the electric potential and the electric field vector economically in order to apply the adjoint state of the poisson equation in the resistivity inverse problem we introduce an apparent electric field defined as the dot product between the computed electric field vector and a weighting factor and then defining a virtual source to compute the partial derivative of the electric field vector we exploit the adjoint state the symmetry of green s function of matrix equations derived from solving the poisson equation by the mixed finite element method for the calculation of the steepest descent direction of our objective function by computing the steepest descent direction by a dot product of backpropagated residual and virtual source we can avoid the cumbersome and expensive process of computing the jacobian matrix directly we calibrate our algorithm on a synthetic of a buried conductive block and obtain an image that is compatible with the limits of the resistivity method

probing for electrical inclusions with complex spherical waves let a physical body in 2 or 3 be given assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background further assume that a subset is available for boundary measurements it is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on more precisely given a ball b with center outside the convex hull of and satisfying b boundary measurements on with explicitly given dirichlet data are enough to determine whether b intersects the inclusion an approximate detection algorithm is introduced based on the theory numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion free domain near and is robust against measurement noise 2007 wiley periodicals inc

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

overdetermined partial boundary value problems on finite networks abstract in this study we define a class of non self adjoint boundary value problems on finite networks associated with schrodinger operators the novel feature of this study is that no data are prescribed on part of the boundary whereas both the values of the function and of its normal derivative are given on another part of the boundary we show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm we analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial dirichlet to neumann maps these maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed we also execute full conductance recovery for spider networks

a direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

an explicit update scheme for inverse parameter and interface estimation of piecewise constant coefficients in linear elliptic pdes we introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities for this purpose we use an output least squares approach with level set and augmented lagrangian methods our formulation incorporates the inherent nature of the piecewise constant coefficients which eliminates the need for a complicated nonlinear solve at every iteration instead we obtain an explicit update formula and therefore vastly speed up computation we employ our approach to the example problems of poisson s equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface

on the hp adaptive solution of complete electrode model forward problems of electrical impedance tomography electrical impedance tomography is a noninvasive imaging technique based on measurements of currents and voltages on the boundary of the object of interest the most accurate forward model for impedance tomography is the complete electrode model that takes into account the electrode shapes and the contact impedances at the corresponding interfaces many practical reconstruction algorithms of electrical impedance tomography require repetitive computation of accurate forward solutions for the complete model in this work we introduce an hp adaptive finite element method for solving the complete electrode forward problem and demonstrate its functionality via two dimensional numerical studies

sparsity reconstruction in electrical impedance tomography we investigate the potential of sparsity constraints in the electrical impedance tomography eit inverse problem of inferring the distributed conductivity based on boundary potential measurements in sparsity reconstruction inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis this prior information is incorporated into a tikhonov type functional by including a sparsity promoting 1 penalty term the functional is minimized with an iterative soft shrinkage type algorithm in this paper the feasibility of the sparsity reconstruction approach is evaluated by experimental data from water tank measurements the reconstructions are computed both with sparsity constraints and with a more conventional smoothness regularization approach the results verify that the adoption of 1 type constraints can enhance the quality of eit reconstructions in most of the test cases the reconstructions with sparsity constraints are both qualitatively and quantitatively more feasible than that with the smoothness constraint

electrical impedance tomography problem with inaccurately known boundary and contact impedances in electrical impedance tomography eit electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body the resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data all the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode skin contact impedances are known a priori however in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances and therefore approximate model domain and contact impedances have to be used in the image reconstruction however it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images leading to loss of diagnostically relevant information in a recent paper kolehmainen 2006 we showed how in the 2 d case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of teichmuller mappings in this paper the theory and reconstruction method are extended to include the estimation of unknown contact impedances the method is implemented numerically and tested with experimental eit data the results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method

reconstruction of permittivity profile from boundary capacitance data we present numerical methods of reconstructing permittivity profile of material flow from data collected in electrical capacitance tomography ect based on a formula for the sensitivity maps from a partial differential equation model we propose a unified framework of iterative algorithms using updated sensitivity maps to properly incorporate the non linear nature of the problem and a regularization mechanism to address the ill posedness for two and three phase flows the total variation tv regularization method is shown to be effective in reconstructing the permittivity profiles accurately in a few iterations without any a priori knowledge of the unknown permittivity values

a multiscale method for model order reduction in pde parameter estimation abstract estimating parameters of partial differential equations pdes is of interest in a number of applications such as geophysical and medical imaging parameter estimation is commonly phrased as a pde constrained optimization problem that can be solved iteratively using gradient based optimization a computational bottleneck in such approaches is that the underlying pdes need to be solved numerous times before the model is reconstructed with sufficient accuracy one way to reduce this computational burden is by using model order reduction mor techniques such as the multiscale finite volume method msfv in this paper we apply msfv for solving high dimensional parameter estimation problems given a finite volume discretization of the pde on a fine mesh the msfv method reduces the problem size by computing a parameter dependent projection onto a nested coarse mesh a novelty in our work is the integration of msfv into a pde constrained optimization framework which updates the reduced space in each iteration we present a computationally tractable way of explicitly differentiating the mor solution that acknowledges the change of basis as we demonstrate in our numerical experiments our method leads to computational savings for large scale parameter estimation problems where iterative pde solvers are necessary and offers potential for additional speed ups through parallel implementation

estimateur d erreur a posteriori hierarchique application aux elements finis mixtes on propose une generalisation de l estimateur a posteriori hierarchique de bank weiser aux formulation mixtes dans le cas d approximations conformes et non conformes avec ou sans integration numerique on presente comme exemples d application la formulation mixte duale du probleme de dirichlet pour le laplacien avec et sans integration numerique

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

a fidelity embedded regularization method for robust electrical impedance tomography electrical impedance tomography eit provides functional images of an electrical conductivity distribution inside the human body since the 1980s many potential clinical applications have arisen using inexpensive portable eit devices eit acquires multiple trans impedance measurements across the body from an array of surface electrodes around a chosen imaging slice the conductivity image reconstruction from the measured data is a fundamentally ill posed inverse problem notoriously vulnerable to measurement noise and artifacts most available methods invert the ill conditioned sensitivity or the jacobian matrix using a regularized least squares data fitting technique their performances rely on the regularization parameter which controls the trade off between fidelity and robustness for clinical applications of eit it would be desirable to develop a method achieving consistent performance over various uncertain data regardless of the choice of the regularization parameter based on the analysis of the structure of the jacobian matrix we propose a fidelity embedded regularization fer method and a motion artifact reduction filter incorporating the jacobian matrix in the regularization process the new fer method with the motion artifact reduction filter offers stable reconstructions of high fidelity images from noisy data by taking a very large regularization parameter value the proposed method showed practical merits in experimental studies of chest eit imaging

electrical impedance tomography for piecewise constant domains using boundary element shape based inverse solutions shape based solutions have recently received attention for certain ill posed inverse problems their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns especially important for nonlinear ill posed problems we apply the shape based approach to current injection electrical impedance tomography eit reconstructions we employ a boundary element method bem based solution for eit we introduce two shape models one based on modified b splines and the other based on spherical harmonics for bem modeling of shapes these methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume we assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not we also assume the internal conductivity profile is piecewise constant meaning that each region has a constant conductivity we propose and test three different regularization techniques to be used with either of the shape models the performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object we observe that in the noisy environment either simulated noise or real sources of noise in the experimental study we get reasonable reconstructions since the signal to noise ratio snr expected in modern eit instruments is higher than that used in this study these reconstruction methods may prove useful in practice

a posteriori error estimate and convergence analysis for conductivity image reconstruction in mreit magnetic resonance electrical impedance tomography mreit takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object when we inject current into the imaging object there occurs a distribution of internal magnetic flux density mathbf b b x b y b z in mreit we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire b z data the harmonic b z algorithm was invented in 2001 to reconstruct cross sectional conductivity images from b z data sets subject to multiple injection currents utilizing internal b z data it overcomes the inherent ill posedness in electrical impedance tomography we can set up the inverse problem in mreit as a coefficient identification problem of finding sigma appearing in nabla cdot sigma nabla u 0 from acquired data of the z component of nabla times sigma nabla u the harmonic b z algorithm has shown an excellent perfor

the neumann to dirichlet map in two dimensions abstract for the two dimensional schrodinger equation in a bounded domain we prove uniqueness of the determination of potentials in w p 1 p 2 in the case where we apply all possible neumann data supported on an arbitrarily non empty open set of the boundary and observe the corresponding dirichlet data on an immediate consequence is that one can uniquely determine a conductivity in w p 3 with p 2 by measuring the voltage on an open subset of the boundary corresponding to a current supported in the same set

electrical impedance tomography by elastic deformation this paper presents a new algorithm for conductivity imaging our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations using asymptotics of the fields in the presence of small volume inclusions we relate the pointwise values of the energy density to the measured data through a nonlinear pde our algorithm is based on this pde and takes full advantage of the enriched data we give numerical examples that illustrate the performance and the accuracy of our approach

convergence of a reconstruction method for the inverse conductivity problem the inverse conductivity problem is that of reconstructing a spatially varying isotropic conductivity in the interior of some region by means of steady state measurements taken at the boundary reconstruction schemes including least squares type minimization methods have been widely studied and implemented but convergence analysis has been largely ignored this paper establishes the convergence of a well known least squares minimization scheme the levenberg marquardt method on a regularized formulation of the inverse conductivity problem

periodic solutions to a second order nonlinear neutral functional differential equation in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t cx t f x t x t g t x t t e t in the critical case c 1 by employing mawhin s continuation theorem and some analysis techniques sufficient conditions are given for the existence of periodic solutions

statistical estimation of resistance conductance by electrical impedance tomography measurements this paper is built upon the assumption that in electrical impedance tomography vectors of voltages and currents are linearly dependent through a resistance matrix this linear relationship was confirmed experimentally and may be derived analytically under certain assumptions regarding electrodes isaacson 1991 given measurement data consisting of voltages and currents we treat this relationship as a linear statistical model thus our goal is not to reconstruct the image but directly estimate its electromagnetic properties reflected in the resistance and or conductance matrix using electrical impedance tomography eit measurements of voltages and currents on the periphery of the body since no inverse problem is involved the algorithm for estimation merely reduces to one matrix inversion we estimate the impedance resistance matrix using well established statistical inference techniques for linear regression models we provide a comprehensive treatment for a two dimensional homogeneous body of a circular shape by which many concepts of electrical impedance tomography such as width of electrodes the difference between voltage current and current voltage systems are illustrated our theory may be applied to various tests including eit hardware calibration and whether the medium is homogeneous these tests are illustrated on phantom agar data

feasibility of simplified integral equation modeling of low frequency marine csem with a resistive target we have assessed the accuracy of a simplified integral equation sie modeling approach for marine controlled source electromagnetics csem with low applied frequencies and a resistive target the most computationally intensive part of rigorous integral equation ie modeling is the computation of the anomalous electric field within the target itself this leads to a matrix problem with a dense coefficient matrix it is well known that in general the presence of many grid cells creates a computational disadvantage for dense matrix methods compared to sparse matrix methods the sie approach replaces the dense matrix part of rigorous ie modeling by sparse matrix calculations based on an approximation of maxwell s equations the approximation is justified theoretically if a certain dimensionless parameter is small as opposed to approximations of the born type the validity of the sie approach does not rely on the anomalous field to be small in comparison with the background field in the target region

optimal experimental design for inverse problems with state constraints in this paper we address the challenging problem of optimal experimental design oed of inverse problems with state constraints we consider two oed formulations that allow us to reduce the exper

identifying conductivity in electrical impedance tomography with total variation regularization in this paper we investigate the problem of identifying the conductivity in electrical impedance tomography from one boundary measurement a variational method with total variation regularization is here proposed to tackle this problem we discretize the pde as well as the conductivity with piecewise linear continuous finite elements we prove the stability and convergence of this technique for the numerical solution we propose a projected armijo algorithm finally a numerical experiment is presented to illustrate our theoretical results

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

comment on calderon s paper on an inverse boundary value problem calder6n determined a method to approximate the conductivity a of a con ducting body in rn for n 2 based on measurements of boundary data the approximation is good in the l norm provided that the conductivity is a small perturbation from a constant we calculate the approximation exactly for the case of homogeneous concentric conducting disks in r2 with different conductivities here the difference in the conductivities is the perturbation we show that the approximation yields precise information about the spatial variation of a even when the perturbation is large this ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

the inverse problem of resistivity sounding the electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two layer systems this new representation is called the bilayer expansion for the green s function it enables the forward problem of resistivity sounding to be solved very efficiently even for complicated profiles also the bilayer expansion facilitates the solution of the resistivity inverse problem the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model specifically i consider models comprising uniformly conducting layers of equal thickness for a given finite data set a quadratic program can be used to find the best fitting model in this class for any specified thickness as the thickness is reduced models of this kind can approximate arbitrary profiles with unlimited accuracy if there is a model that satisfies

multifrequency trans admittance scanner mathematical framework and feasibility a trans admittance scanner tas is a device for breast cancer diagnosis based on numerous experimental findings that complex conductivities of breast tumors significantly differ from those of surrounding normal tissues in tas we apply a sinusoidal voltage between a hand held electrode and a scanning probe placed on the breast skin to make current travel through the breast the scanning probe has an array of electrodes at zero voltage we measure exit currents neumann data through the electrodes that provide a map of trans admittance data over the breast surface the inverse problem of tas is to detect a suspicious abnormality underneath the breast skin from the measured neumann data previous anomaly detection methods used the difference between the measured neumann data and a reference neumann data obtained beforehand in the absence of anomaly however in practice the reference data is not available and its computation is not possible since the inhomogeneous complex conductivity of the normal breast is unknown to deal with this problem we propose a frequency difference tas fdtas in which a weighted frequency difference of the trans admittance data measured at a certain moment is used for anomaly detection this paper provides a mathematical framework and the feasibility of fdtas by showing the relationship between the anomaly information and the weighted frequency difference of the neumann data

an alternating direction algorithm for total variation reconstruction of distributed parameters augmented lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems the alternating direction method of multipliers admm is one of such formulations optimization methods very recently the number of applications of the admm or variants of it to solve inverse problems in image and signal processing has increased at an exponential rate the reason for this interest is that admm decomposes a difficult optimization problem into a sequence of much simpler problems in this paper we use the admm to reconstruct piecewise smooth distributed parameters of elliptical partial differential equations from noisy and linear blurred observations of the underlying field the distributed parameters are estimated by solving an inverse problem with total variation tv regularization the proposed instance of the admm solves in each iteration an and a decoupled optimization problems an operator splitting is used to simplify the treatment of the tv regularizer avoiding its smooth approximation and yielding a simple yet effective admm reconstruction method compared with previously proposed approaches the competitiveness of the proposed method with respect to the state of the art is illustrated in simulated 1 d and 2 d elliptical equation problems which are representative of many real applications

first order system least squares and electrical impedance tomography electrical impedance tomography eit belongs to a family of imaging methods that employ boundary measurements to distinguish interior spatial variation of an electromagnetic parameter the associated inverse problem is notoriously ill posed due to diffusive effects in the quasi static regime when electrical impedance reduces to its real part resistivity the standard approach to eit is output least squares ols for a set of applied normal boundary currents one minimizes the defect between the measured and computed boundary voltages associated respectively with the exact impedance and its approximation in minimizing a boundary functional ols implicitly imposes the governing poisson equation as an optimization constraint to reconstruct resistivity or equivalently conductivity we introduce a new first order system least squares fosls formulation that incorporates the elliptic pde as an interior functional in a global unconstrained minimization scheme we then establish equivalence of our functional to ols and to an existing interior least squares functional due to kohn and vogelius comm pure appl math 37 1984 pp 289 298 that the latter may be viewed as a special dual approach fosll is an interesting attribute of this equivalence because the fosls functional implicitly reflects the inherent loss of resolution away from the boundary relative to the set of applied boundary tests the need for artificial regularization may be avoided

projective electrical impedance reconstruction with two measurements this paper presents a projective reconstruction method of electrical impedance distribution using two boundary measurements of current voltage data the proposed method has a major advantage over existing electrical impedance tomography systems relatively high resolution impedance image in a near surface region underneath the voltage sensing probe that is placed on a skin surface we attach two pairs of electrodes on the border of the probe in such a way that each pair of electrodes generates electrical current flowing underneath the voltage sensing probe and two currents flow across each other a careful analysis of the measured data with respect to local perturbation of admittivity leads to the projective reconstruction method and its rationale numerical simulations show that the proposed method successfully reconstructs high resolution image of local perturbation of admittivity due to object position changes or admittivity value changes or the both in a near surface region underneath the probe

convergence and regularization for monotonicity based shape reconstruction in electrical impedance tomography the inverse problem of electrical impedance tomography is severely ill posed meaning that only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage recently it was shown that a simple monotonicity property of the related neumann to dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity in this paper we formulate a monotonicity based shape reconstruction scheme that applies to approximative measurement models and regularizes against noise and modelling error we demonstrate that for admissible choices of regularization parameters the inhomogeneities are detected and under reasonable assumptions asymptotically exactly characterized moreover we rigorously associate this result with the complete electrode model and describe how a computationally cheap monotonicity based reconstruction algorithm can be implemented numerical reconstructions from both simulated and real life measurement data are presented

nonlinear fourier analysis for discontinuous conductivities computational results abstract two reconstruction methods of electrical impedance tomography eit are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics cgo solutions to d bar equations involving the global uniqueness proofs for calderon problem exposed in nachman 1996 43 and astala and paivarinta 2006 6 the astala paivarinta theory based low pass transport matrix method implemented in astala et al 2011 3 and the shortcut method which considers ingredients of both theories the latter method is formally similar to the nachman theory based regularized eit reconstruction algorithm studied in knudsen et al 2009 34 and several references from there new numerical results are presented using parallel computation with size parameters larger than ever leading mainly to two conclusions as follows first both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases and there seems to be a gibbs like phenomenon producing ringing artifacts second the transport matrix method loses accuracy away from a freely chosen pivot point located outside of the object to be studied whereas the shortcut method produces reconstructions with more uniform quality

overdetermined partial resolvent kernels for finite networks abstract in 2 a study of the existence and uniqueness of solution of partial overdetermined boundary value problems for finite networks was performed these problems involve schrodinger operators and the novel feature is that no data are prescribed on part of the boundary whereas both the values of the function and of its normal derivative are given on another part of the boundary in the present work we study the resolvent kernels associated with overdetermined partial boundary value problems on finite network and we express them in terms of the well known green operator and the dirichlet to robin map moreover we analyze their main properties and we compute them in the case of a generalized cylinder the obtained expression involve polynomials that can be seen as a generalization of chebyshev polynomials and indeed when the conductances along axes are constant the expressions for the overdetermined partial resolvent kernels are given in terms of second kind chebyshev polynomials

domain reconstruction using photothermal techniques a numerical method to detect objects buried in a medium by surface thermal measurements is presented we propose a new approach combining the use of topological derivatives and laplace transforms the original optimization problem with time dependent constraints is replaced by an equivalent problem with stationary constraints by means of laplace transforms the first step in the reconstruction scheme consists in discretizing the inversion formula to produce an approximate optimization problem with a finite set of constraints then an explicit expression for the topological derivative of the approximate shape functional is given this formula is evaluated at low cost using explicit expressions of the forward and adjoint fields involved we apply this technique to a simple shape reconstruction problem set in a half space good approximations of the number location and size of the obstacles are obtained the description of their shapes can be improved by more expensive hybrid methods combining time averaging with topological derivative based iterative schemes

a partially reflecting random walk on spheres algorithm for electrical impedance tomography in this work we develop a probabilistic estimator for the voltage to current map arising in electrical impedance tomography this novel so called partially reflecting random walk on spheres estimator enables monte carlo methods to compute the voltage to current map in an embarrassingly parallel manner which is an important issue with regard to the corresponding inverse problem our method uses the well known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions we analyze the global bias and the variance of the new estimator both theoretically and experimentally subsequently the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms

electrical impedance tomography using level set representation and total variational regularization in this paper we propose a numerical scheme for the identification of piecewise constant conductivity coefficient for a problem arising from electrical impedance tomography the key feature of the scheme is the use of level set method for the representation of interface between domains with different values of coefficients numerical tests show that our method can recover sharp interfaces and can tolerate a relatively high level of noise in the observation data results concerning the effects of number of measurements noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given

existence and uniqueness for electrode models for electric current computed tomography the following experiment is considered to a body of given conductivity a number of electrodes are attached through which current is sent on the same electrodes the resulting voltages are measured this experiment can be described by a number of mathematical models k s cheng et al ieee transactions on biomedical engineering 36 1989 pp 918 924 these models are discussed and their predictions compared with experiment in particular a model is exhibited that is capable of predicting the experimentally measured voltages to within 0 1 percent for this model existence and uniqueness of the associated electrical potential is proved

on the continuum limit of a discrete inverse spectral problem on optimal finite difference grids we consider finite difference approximations of solutions of inverse stur m liouville problems in bounded intervals using three point finite difference sc hemes we discretize the equations on so called optimal grids constructed as follow s for a staggered grid with 2k points we ask that the finite difference operator a k k jacobi matrix and the sturm liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval this requirement determines uniquely the entries in the jacobi matrix which are grid cell averages of the coefficients in the continuum problem if these coefficients are known we can find the grid which we call optimal becau se it gives by design a finite difference operator with a prescribed spectr al measure we focus attention on the inverse problem where neither the coefficients n or the grid are known a key question in inversion is how to parametrize the coefficients i e how to choose the grid it is clear that to be successful this grid must be close to the optimal one which is unknown fortunately as we show here the grid dependence on the unknown coefficients is weak so the inversion can be done on a pr ecomputed grid for an a priori guess of the unknown coefficients this obse rvation leads to a simple yet efficient inversion algorithm which gives coefficien ts that converge pointwise to the true solution as the number k of data points tends to infinity the cornerstone of our convergence proof is showing that op timal grids provide an implicit natural regularization of the inverse problem by givin g reconstructions with uniformly bounded total variation the analysis is based on a novel explicit perturbation analysis of lanczos recursions and on a discrete