Table of contents

In this paper, a Fisher information analysis is employed to establish some important physical performance bounds in microwave tomography. As a canonical problem, the two-dimensional electromagnetic inverse problem of imaging a cylinder with isotropic dielectric losses is considered. A fixed resolution is analysed by introducing a finite basis, i.e., pixels representing the material properties. The corresponding Cramér–Rao bound for estimating the pixel values is computed based on a calculation of the sensitivity field which is obtained by differentiating the observed field with respect to the estimated parameter. An optimum trade-off between the accuracy and the resolution is defined based on the Cramér–Rao bound, and its application to assess a practical resolution limit in the inverse problem is discussed. Numerical examples are included to illustrate how the Fisher information analysis can be used to investigate the significance of measurement distance, operating frequency and losses in the canonical tomography set-up.

A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg–Marquardt methods, a feasible-path approach and an sequential quadratic programming-type method, are analysed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

We study an inverse spectral problem for arbitrary order ordinary differential equations on compact star-type graphs. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are a generalization of the Weyl function (m-function) for the classical Sturm–Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

We develop direct and inverse scattering theory for Jacobi operators with a steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand–Levitan–Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.

In this paper we study the inverse scattering problem and the inverse boundary value problem for the Dirac operator with compact supported potential. We prove a 'gauge equation' for two potentials of arbitrary forms when their corresponding scattering amplitudes at fixed energy or Dirichlet-to-Dirichlet maps are the same. We also use this equation to determine the generalized magnetic potential up to a gauge equivalent class.

In this paper, we present a method for solving inverse Sturm–Liouville problems by generalizing a Rundell–Sacks algorithm. The method is extended to deal with a general reference potential which can be adapted, e.g., to estimations of the jump-discontinuity points of the unknown potential. Moreover, its convergence properties are investigated. Numerical examples show that this modification can achieve more precise results from a given data set than the earlier method in using only the null reference potential and, therefore, the L²- and L^∞-error can be reduced significantly.

The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the ℓ¹ case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.

We consider the problem of reconstructing a sparse signal from a limited number of linear measurements. Given m randomly selected samples of Ux⁰, where U is an orthonormal matrix, we show that ℓ₁ minimization recovers x⁰ exactly when the number of measurements exceeds where S is the number of nonzero components in x⁰ and μ is the largest entry in U properly normalized: . The smaller μ is, the fewer samples needed. The result holds for 'most' sparse signals x⁰ supported on a fixed (but arbitrary) set T. Given T, if the sign of x⁰ for each nonzero entry on T and the observed values of Ux⁰ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about as many samples.

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

In this paper, we investigate the convergence behaviour of the Krasnoselski–Mann (KM) iteration and its generalizations. The KM iteration may be written as follows: where N is a nonexpansive operator on a Hilbert space . This scheme aims to find fixed points of the operator N. Many problems from various fields, including the inverse problems area, can be expressed as a fixed point problem of a certain operator N. In earlier articles, the convergence of the KM iteration and its generations have been investigated in the case when the operator N is nonexpansive and has the fixed points. This paper further studies the convergence behaviour of the algorithms discussed. We first extend the convergence result for the KM iteration to the case when the operator N is firmly nonexpansive, in which case the relaxation parameters are allowed to be in the interval [0, 2], instead of [0, 1]. Then, we show that this result remains valid for the generalized KM iterations. Furthermore, we prove that the sequences generated from the KM iteration or the generalized KM iterations are unbounded if the operators related to the iterations have no fixed points.

This paper proposes a nonlinear 3D deformable model for the image segmentation of soft structures. The template is modelled as an elastic body which is deformed by forces derived from the image. It relies on a template, which is a topological, geometrical and material model of the structure to segment. This model is based on the nonlinear three-dimensional elasticity problem with a boundary condition of pure traction. In addition, the applied forces are successive, as they depend on the displacements. For computations, an incremental algorithm is proposed to minimize the global energy of template deformation. Sufficient conditions of the convergence for the incremental algorithm are given. Finally, a discrete algorithm using the finite element method is presented and evaluated on synthetic images and actual MR images of mouse hearts.

We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such a question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependence theory for the solution in terms of the coefficients and, secondly, a novel regularization technique for tackling this inverse problem which takes into account the specific nature of the equation. Our results also rely on generalized relative entropy estimates and related Poincaré inequalities.

We consider a two-dimensional inverse heat conduction problem for a slab. This is a severely ill-posed problem. Two regularization strategies, one based on the modification of the equation, the other based on the truncation of high frequency components, are proposed to solve the problem in the presence of noisy data. Error estimates show that the regularized solution is dependent continuously on the data and is an approximation of the exact solution of the two-dimensional inverse heat conduction problem. The relation of these two and other regularization strategies is also discussed.

There are industrial applications where the recovery of the coefficients of the heat conduction equation from measurements of the temperature over an open set Ω* is crucial. We analyse the inverse problem of identifying the conductivity coefficient of the heat equation when a zero initial condition is set and single measurements are made. We prove a uniqueness result for a linearized version of this problem in for n odd that does not depend on a hypothesis about the relative position of the support of the unknown function with respect to Ω*. It is an extension, for n odd, of a theorem proved by Elayyan and Isakov.

In this paper, we describe a new and simple method for the reconstruction of the shape of a perfectly conducting object illuminated by a single plane wave at a fixed frequency. The basic idea of the method is to use the condition that the field must vanish on the (unknown) contour together with convenient representations of the scattered field. In particular, by means of a regularized single-layer potential approach, the measured data are first analytically continued to a circle closely covering the object, while a Taylor expansion in the radial direction is exploited to represent the field in the vicinity of the target. From the boundary condition, the problem is then recast as a polynomial equation containing the contour of the object as an unknown. This nonlinear equation is iteratively solved via the Newton method and it is regularized using the method of least squares. As shown by several numerical examples, the proposed method is computationally effective, it is robust against uncertainties on data and, despite the very limited number of data which are exploited, yields satisfactory reconstructions for convex and concave-shaped star-like scatterers of size comparable to the wavelength.

Nondestructive evaluation of hidden surface damage by means of stationary thermographic methods requires the construction of approximated solutions of a boundary identification problem for an elliptic equation. In this paper, we describe and test a regularized reconstruction algorithm based on the linearization of this class of inverse problems. The problem is reduced to an infinite linear system whose coefficients come from the Fourier discretization of the Robin boundary value problem for Laplace's equation.

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.

In part I (Guven et al2007 Inverse Problems23 1115–33), we analysed the error in the reconstructed optical absorption images resulting from the discretization of the forward and inverse problems. Our analysis led to two new error estimates, which present the relationship between the optical absorption imaging accuracy and the discretization error in the solutions of the forward and inverse problems. In this work, based on the analysis presented in part I, we develop new adaptive discretization schemes for the forward and inverse problems in order to reduce the error in the reconstructed images resulting from discretization. The proposed discretization schemes lead to adaptively refined composite meshes that yield the desired level of imaging accuracy while reducing the size of the discretized forward and inverse problems. We present numerical experiments to validate the error estimates developed in part I and show the improvement in the accuracy of the reconstructed optical images with the new adaptive mesh generation algorithms.

We consider a partially coated anisotropic medium and determine its shape by knowledge of Cauchy data on the boundary of a domain where we a priori know that the obstacle is located. We then give a variational characterization of the supreme of the surface conductivity and validate the method with some numerical examples.

We study an inverse spectral problem of reconstructing a class of oscillating systems consisting of a continuous part coupled with a discrete part with N degrees of freedom. The operators corresponding to these systems act in and are composed of Sturm–Liouville operators in L₂(0, 1) with distributional potentials and Jacobi matrices in , coupled in a special way. We solve the inverse spectral problem for the operators and describe explicitly the set of their spectral data. We also exhibit a connection to related Sturm–Liouville problems with boundary conditions depending rationally on the spectral parameter.

In this paper, the topic of optimal experiment design in electrical impedance tomography (EIT) is studied. More specifically, we consider determination of optimal current patterns in EIT in cases of time-varying targets. The reconstruction problem associated with EIT imaging is known to be an ill-posed inverse problem. Statistical inversion methods have been shown to be advantageous in many cases in EIT. In Kaipio et al (2004 Inverse Problems20 919–36), we considered the problem of optimal experiment design in statistical framework and we proposed an approach for determining optimal current patterns in cases of imaging of time-invariant targets. The approach was based on the statistical interpretation of the reconstruction problem and optimal current patterns were obtained by minimizing the trace of an approximate posterior covariance matrix. In this paper, we utilize a similar approach to determining optimal current patterns in cases of time-varying targets. The image reconstruction problem of EIT is formulated as a state estimation problem. As in the time-invariant case, the optimality criterion is based on the posterior covariances but instead of considering one specific time instant we minimize the time-averaged mean posterior variance. It is shown in a numerical study that the uncertainties of the estimates obtained with optimized current patterns are smaller than those obtained with conventional current patterns. In addition, the results indicate that in time-varying problems a single optimized current pattern may be sufficient to achieve good accuracy, i.e., multiple optimized current patterns do not provide substantial further information on the target. We also demonstrate that in some cases the increase in the number of current patterns can even decrease the reliability of the estimates. This is one of the reasons for the topic of optimal current patterns being quite important in the case of imaging time-varying targets.

Bioluminescence tomography (BLT) is a rapidly developing new area of molecular imaging. The goal of BLT is to produce a quantitative reconstruction of a bioluminescent source distribution within a living mouse from bioluminescent signals measured on the body surface of the mouse. While in most BLT studies so far the optical parameters of the key anatomical regions are assumed known from the literature or diffuse optical tomography (DOT), these parameters cannot be very accurate in general. In this paper, we propose and study a new BLT approach that optimizes optical parameters when an underlying bioluminescent source distribution is reconstructed to match the measured data. We prove the solution existence and the convergence of numerical methods. Also, we present numerical results to illustrate the utility of our approach and evaluate its performance.

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation , known for every geodesic γ of a given Riemannian metric. Here, is the orthogonal projection onto the hyperplane . The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a C¹-small function f can be recovered from the data uniquely up to a natural obstruction. A partial global result is obtained in the case of the Euclidean metric on .

This work deals with the inverse Born approximation for the nonlinear two-dimensional Schrödinger equation with cubic nonlinearity where the real-valued unknown functions q and α belong to with some behaviour at infinity. The following problem is studied: to estimate the smoothness of the terms from the Born sequence which corresponds to the scattering data with all arbitrary large energies and all angles in the scattering amplitude. These smoothness estimates allow us to conclude that the leading order singularities of the sum of unknown functions q and α can be obtained exactly by the Born approximation. Especially, we show for the functions in L^p, for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces. In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by these scattering data.

This paper considers inverse problems of shape recovery from noisy boundary data, where the forward problem involves the inversion of elliptic PDEs. The piecewise constant solution, a scaling and translation of a characteristic function, is described in terms of a smoother level set function. A fast and simple dynamic regularization method has been recently proposed that has a robust stopping criterion and typically terminates after very few iterations. Direct linear algebra methods have been used for the linear systems arising in both forward and inverse problems, which is suitable for problems of moderate size in 2D. For larger problems, especially in 3D, iterative methods are required. In this paper we extend our previous results to large-scale problems by proposing and investigating iterative linear system solvers in the present context. Perhaps contrary to one's initial intuition, the iterative methods are particularly useful for the inverse rather than the forward linear systems. Moreover, only very few preconditioned conjugate gradient iterations are applied towards the solution of the linear system for the inverse problem, allowing the regularizing effects of such iterations to take centre stage. The efficacy of the obtained method is demonstrated.

A mathematical formulation that provides a converging solution for the successive approximation solution of the inverse problem of imaging with incoherently scattered radiation is introduced. The nonlinear nature of this problem can cause its solution to oscillate between two physically acceptable domains. By nonlinear scaling of the forward problem, it is shown that the range of one of the two domains can be made to expand at the expense of the other. This enables contraction mapping of the iterative solution of the inverse problem over an extended domain. The mathematical features of this scaling approach are analytically demonstrated for a one-pixel inverse problem, to elucidate its features and limitations. Reconstruction for many-pixel tomographic images is then presented for ideal (error free) and noisy simulated measurements, demonstrating the ability of the presented scheme to solve the inverse problem for radiation scatter imaging over a wide range of physically acceptable attributes.

We are concerned with a problem arising in corrosion detection. We consider the stability issue for the inverse problem of determining a Robin coefficient on the inaccessible portion of the boundary by the electrostatic measurements performed on the accessible one. We provide a Lipschitz stability estimate under the further a priori assumption of a piecewise constant Robin coefficient. Furthermore, we prove that the Lipschitz constant of the above-mentioned estimate behaves exponentially with respect to the number of the portions considered.

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