Showing papers in "Inverse Problems in 2000"

Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography

Abstract: This paper discusses the electrical impedance tomography (EIT) problem: electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes. The corresponding voltages that are needed to maintain these currents are measured. The objective is to estimate the unknown resistivity, or more generally the impedivity distribution of the body based on this information. The most commonly used method to tackle this problem in practice is to use gradient-based local linearizations. We give a proof for the differentiability of the electrode boundary data with respect to the resistivity distribution and the contact impedances. Due to the ill-posedness of the problem, regularization has to be employed. In this paper, we consider the EIT problem in the framework of Bayesian statistics, where the inverse problem is recast into a form of statistical inference. The problem is to estimate the posterior distribution of the unknown parameters conditioned on measurement data. From the posterior density, various estimates for the resistivity distribution can be calculated as well as a posteriori uncertainties. The search of the maximum a posteriori estimate is typically an optimization problem, while the conditional expectation is computed by integrating the variable with respect to the posterior probability distribution. In practice, especially when the dimension of the parameter space is large, this integration must be done by Monte Carlo methods such as the Markov chain Monte Carlo (MCMC) integration. These methods can also be used for calculation of a posteriori uncertainties for the estimators. In this paper, we concentrate on MCMC integration methods. In particular, we demonstrate by numerical examples the statistical approach when the prior densities are non-differentiable, such as the prior penalizing the total variation or the L1 norm of the resistivity.

On optimization techniques for solving nonlinear inverse problems

Abstract: This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilizing a Gauss-Newton method in which the forward model constraints are implicitly incorporated. Variants of Newton's method which use second-derivative information are rarely employed because their perceived disadvantage in computational cost per step offsets their potential benefits of faster convergence. In this paper we show that, by formulating the inversion as a constrained or unconstrained optimization problem, and by employing sparse matrix techniques, we can carry out variants of sequential quadratic programming and the full Newton iteration with only a modest additional cost. By working with the differential equation explicitly we are able to relate the constrained and the unconstrained formulations and discuss the advantages of each. To make the comparisons meaningful we adopt the same global optimization strategy for all inversions. As an illustration, we focus upon a 1D electromagnetic (EM) example simulating a magnetotelluric survey. This problem is sufficiently rich that it illuminates most of the computational complexities that are prevalent in multi-source inverse problems and we therefore describe its solution process in detail. The numerical results illustrate that variants of Newton's method which utilize second-derivative information can produce a solution in fewer iterations and, in some cases where the data contain significant noise, requiring fewer floating point operations than Gauss-Newton techniques. Although further research is required, we believe that the variants proposed here will have a significant impact on developing practical solutions to large-scale 3D EM inverse problems.

Prolate Spheroidal Wave Functions, Quadrature, and Interpolation

Abstract: Polynomials are one of the principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc, it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well developed apparatus of prolate spheroidal wavefunctions to construct quadratures, interpolation and differentiation formulae, etc, for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc, the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.

A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets

Abstract: A two-step shape reconstruction method for electromagnetic (EM) tomography is presented which uses adjoint fields and level sets. The inhomogeneous background permittivity distribution and the values of the permittivities in some penetrable obstacles are assumed to be known, and the number, sizes, shapes, and locations of these obstacles have to be reconstructed given noisy limited-view EM data. The main application we address in the paper is the imaging and monitoring of pollutant plumes in environmental cleanup sites based on cross-borehole EM data. The first step of the reconstruction scheme makes use of an inverse scattering solver which recovers equivalent scattering sources for a number of experiments, and then calculates from these an approximation for the permittivity distribution in the medium. The second step uses this result as an initial guess for solving the shape reconstruction problem. A key point in this second step is the fusion of the `level set technique' for representing the shapes of the reconstructed obstacles, and an `adjoint field technique' for solving the nonlinear inverse problem. In each step, a forward and an adjoint Helmholtz problem are solved based on the permittivity distribution which corresponds to the latest best guess for the representing level set function. A correction for this level set function is then calculated directly by combining the results of these two runs. Numerical experiments are presented which show that the derived method is able to recover one or more objects with nontrivial shapes given noisy cross-borehole EM data.

An Inverse Source Problem in Potential Analysis

Abstract: This paper discusses some aspects of an inverse source problem for elliptic equations, with observations on the boundary of the domain. The main application aimed at is the problem of identifying electrostatic dipoles in the human head where the boundary data are collected via electrodes placed on a part of the head. An uniqueness result is established for dipolar sources. Through solving a finite number of Cauchy problems, one arrives at an inverse problem in the homogeneous case. Assuming the number of dipoles bounded by a known integer M, we have established an algorithm which allows us to identify the number, the locations and moments of the dipoles by algebraic considerations. Other types of sources are also considered.

Numerical implementation of two noniterative methods for locating inclusions by impedance tomography

Abstract: Electrical impedance tomography is applied to recover inclusions within a body from electrostatic measurements on the surface of the body. Here, an inclusion is defined to be a region where the electrical conductivity differs significantly from the background. Recently, theoretical foundations have been developed for new techniques to localize inclusions from impedance tomography data. In this paper it is shown that these theoretical results lead quite naturally to noniterative numerical reconstruction algorithms. The algorithms are applied to a number of test cases to compare their performance.

An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem

Abstract: The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ??2 from knowledge of the Dirichlet-to-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman's proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast.

One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization

Abstract: In this paper, based on the conditional stability estimate for ill-posed inverse problems, we propose a new strategy for a priori choice of regularizing parameters in Tikhonov's regularization and we show that it can be applied to a wide class of inverse problems. The convergence rate of the regularized solutions is also proved.

Reconstructing planar domains from their moments

Abstract: In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed.

A linear sampling method for inverse scattering from an open arc

Abstract: In this paper, we develop a linear sampling method for the inverse scattering of time-harmonic plane waves by open arcs. We derive a characterization of the scatterer in terms of the spectral data of the scattering matrix analogously to the case of the scattering by bounded open domains. Numerical examples show that this theoretical result also leads to a very fast visualization technique for the unknown arc.

A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction

Abstract: In this paper we investigate the stability and convergence rates of the widely used output least-squares method with Tikhonov regularization for the identification of the conductivity distribution in a heat conduction system. Due to the rather restrictive source conditions and regularity assumptions on the nonlinear parameter-to-solution operator concerned, the existing Tikhonov regularization theory for nonlinear inverse problems is difficult to apply for the convergence rate analysis here. By introducing some new techniques, we are able to relax these regularity requirements and derive a much simpler and easily interpretable source condition but still achieve the same convergence rates as the standard Tikhonov regularization theory does.

Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements

Abstract: We consider the 2D inverse conductivity problem for conductivities of the form γ = 1 + χDh defined in a bounded domain Ω⊂2 with C∞ boundary ∂Ω. Here D⊂Ω and hL∞(D) are such that γ has a jump along ∂D. It was shown by Ikehata (Ikehata M J. Inverse and Ill-Posed Problems at press) that the Dirichlet-Neumann map determines the indicator function Iω(τ,t) that can be used to find the convex hull of D. In this paper we find numerically the indicator function for examples with constant h and recover the convex hull of D.

3D electromagnetic inversion based on quasi-analytical approximation

Abstract: In this paper we address one of the most challenging problems of electromagnetic (EM) geophysical methods: three-dimensional (3D) inversion of EM data over inhomogeneous geological formations. The difficulties in the solution of this problem are two-fold. On the one hand, 3D EM forward modelling is an extremely complicated and time-consuming mathematical problem itself. On the other hand, the inversion is an unstable and ambiguous problem. To overcome these difficulties we suggest using, for forward modelling, the new quasi-analytical (QA) approximation developed recently by Zhdanov et al (Zhdanov M S, Dmitriev V I, Fang S and Hursan G 1999 Geophysics at press). It is based on ideas similar to those developed by Habashy et al (Habashy T M, Groom R W and Spies B R 1993 J. Geophys. Res. 98 1759-75) for a localized nonlinear approximation, and by Zhdanov and Fang (Zhdanov M S and Fang S 1996a Geophysics 61 646-65) for a quasi-linear approximation. We assume that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through a scalar electrical reflectivity coefficient, which is a function of the background geoelectrical cross-section and the background EM field only. This approach leads to construction of the QA expressions for an anomalous EM field and for the Frechet derivative operator of a forward problem, which simplifies dramatically the forward modelling and inversion. To obtain a stable solution of a 3D inverse problem we apply the regularization method based on using a focusing stabilizing functional introduced by Portniaguine and Zhdanov (Portniaguine O and Zhdanov M S 1999 Geophysics 64 874-87). This stabilizer helps generate a sharp and focused image of anomalous conductivity distribution. The inversion is based on the re-weighted regularized conjugate gradient method.

On reconstruction in the inverse conductivity problem with one measurement

Abstract: We consider an inverse problem for electrically conductive material occupying a domain ? in 2. Let ? be the conductivity of ?, and D a subdomain of ?. We assume that ? is a positive constant k on D, k?1 and is 1 on ?D; both D and k are unknown. The problem is to find a reconstruction formula of D from the Cauchy data on ?? of a non-constant solution u of the equation ????u = 0 in ?. We prove that if D is known to be a convex polygon such that diamD

Three-dimensional microwave tomography. Theory and computer experiments in scalar approximation

Abstract: The results of computer simulated experiments in three-dimensional microwave tomography in scalar approximation are presented. The gradient method is employed to solve three-dimensional high-contrast microwave tomographic problems. A computer model for full-scale three-dimensional imaging has been created. Three-dimensional tomographic images of mathematical models of the human torso were obtained. Significant differences between two-dimensional and three-dimensional cases are emphasized. Some illumination schemes which can be applied in the three-dimensional case are discussed. A dependence of image quality on the number of vertically placed transmitters has been demonstrated. The computer simulation showed that three-dimensional full-scale human torso dielectrical properties images can be produced with acceptable computational time.

Solution strategies for two- and three-dimensional electromagnetic inverse problems

Abstract: We analyse a variety of solution strategies for nonlinear two-dimensional (2D) and three-dimensional (3D) electromagnetic (EM) inverse problems. To impose a realistic parameterization on the problem, the finite-difference equations arising from Maxwell equations are employed in the forward problem. Krylov subspace methods are then used to solve the resulting linear systems. Because of the efficiencies of the Krylov methods, they are used as the computational kernel for solving 2D and 3D inverse problems, where multiple solutions of the forward problem are required. We derive relations for computing the full Hessian matrix and functional gradient that are needed for computing the model update, via the Newton iteration. Different strategies are then discussed for economically carrying out the Newton iteration for 2D and 3D problems, including the incorporation of constraints necessary to stabilize the inversion process. Two case histories utilizing EM inversion are presented. These include inversion of cross-well data for monitoring electrical conductivity changes arising from an enhanced oil recovery project and the usefulness of cross-well EM methods to characterize the transport pathways for contaminants in the subsurface.

Nondestructive evaluation using a reduced order computational methodology

Abstract: This paper uses eddy current based techniques and reduced order modeling to explore the feasibility of detecting a subsurface damage in structures such as air foils and pipelines. To identify the geometry of a damage, an optimization algorithm is employed which requires solving the forward problem numerous times. To implement these methods in a practical setting, the forward algorithm must be solved with extremely fast and accurate solution methods. Therefore, our computational methods are based on the reduced order Karhunen-Loeve or Proper Orthogonal Decomposition (POD) techniques. For proof-of-concept, we implement the methodology on a 2-D problem and find the methods to be efficient and robust even with data containing 10 Furthermore, the methods are fast; our findings suggest we can reduce the computational time on average by a factor of 3000.

On the equivalence of the Newton-Kantorovich and distorted Born methods

Abstract: We show that the Newton-Kantorovich and distorted Born methods for the computational solution of the nonlinear inverse scattering problem are equivalent This was already shown for the discrete matrix case Here we present an analysis based on the analytic representations of the integral operators We first briefly review both methods and then show that they are equivalent

Design strategies for electromagnetic geophysical surveys

Abstract: Acquiring information on the Earth's electric and magnetic properties is a critical task in many geophysical applications. Since electromagnetic (EM) geophysical methods are based on nonlinear relationships between observed data and subsurface parameters, designing experiments that provide the maximum information content within a given budget can be quite difficult. Using examples from direct-current electrical and frequency-domain EM applications, we review four approaches to quantitative experimental design. Repeated forward modelling is effective in feasibility studies, but may be cumbersome and time-consuming for studying complete data and model spaces. Examining Frechet derivatives provides more insights into sensitivity to perturbations of model parameters, but only in the linear space around the trial model and without easily accounting for combinations of model parameters. A related sensitivity measure, the data importance function, expresses the influence each data point has on determining the final inversion model. It considers simultaneously all model parameters, but provides no information on the relative position of the individual points in the data space. Furthermore, it tends to be biased towards well resolved parts of the model space. Some of the restrictions of these three methods are overcome by the fourth approach, statistical experimental design. This robust survey planning method, which is based on global optimization algorithms, can be customized for individual needs. It can be used to optimize the survey layout for a particular subsurface structure and is an appropriate procedure for nonlinear experimental design in which ranges of subsurface models are considered simultaneously.

Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map

Abstract: We consider the conductivity problem in two dimensions. We show that a complex-valued coefficient , whose imaginary part is small, can be recovered from the knowledge of the Dirichlet-to-Neumann map.

Generalized dual space indicator method for underwater imaging

Abstract: This paper presents a generalized dual space indicator method for imaging an obstacle in ocean environments. The method is based on the observation that the combination (weighted integration) of the measured scattered field can approximate the Green function very well when the Green function's source point is inside the obstacle, but not so well when the source is outside the obstacle. We set up an integral equation whose right-hand side is the Green function with a source point from a searching region. From our numerical experiments, we notice that the norm of the solution of the integral equation has local extrema that lie inside the unknown obstacle. Plotting the norm as a function of the source point in the searching region, and filtering out the region with no local extrema of the norm, we obtain a good image of the unknown obstacle. Imaging algorithms and numerical examples are presented.

On the uniqueness of velocity reconstruction in conducting fluids from measurements of induced electromagnetic fields

Abstract: The problem of velocity reconstruction in conducting fluids from measurements of induced magnetic fields and electric potentials is discussed in spherical geometry. Under the special case that the externally applied magnetic field is uniform and homogeneous throughout the fluid, the non-uniqueness problem is treated in detail. Assuming kinetic energy minimization for the moving fluid, it is shown that the velocity field can be reconstructed completely.

Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems

Abstract: In this paper we study the regularization of linear and nonlinear ill-posed operator equations by projection onto finite-dimensional spaces with a posteriori chosen space dimension. We show that this results in a regularization method, i.e. a stable solution method for the ill-posed problem, that converges to an exact solution as the data noise level goes to zero, with optimal rates under additional regularity conditions.

Locating the support of objects contained in a two-layered background medium in two dimensions

Abstract: In this paper the inverse problem of determining the boundaries of bounded objects buried in a two-layered medium using the regularized sampling method is considered. The objects are taken to be either impenetrable obstacles or penetrable anisotropic scatterers and contained in the absorbing layer of a two-layered medium. Measurements of the scattered field are restricted to the layer not containing the scatterer.

The 3D Doppler transform: elementary properties and computation of reconstruction kernels

Abstract: The 3D Doppler transform maps a vector field to its line integrals over that component of the field which is parallel to the line. In this paper we consider only lines aligned to the coordinate axes. Since the Doppler transform describes the mathematical model for the vector tomography, efficient inversion formulae are necessary in order to solve the reconstruction problem. The approximate inverse represents a numerical inversion scheme based on scalar products of the data with so-called reconstruction kernels. We characterize these reconstruction kernels as solutions of a normal equation connected with the Doppler transform and a mollifier. To solve this equation elementary properties of the underlying operator are investigated and a smoothing property is proved. We succeed in computing a reconstruction kernel for one special mollifier and give a representation with the help of the singular value decomposition of the 2D Radon transform.

Identification of the diffusion coefficient in a one-dimensional parabolic equation

Abstract: This paper deals with the problem of the identification of the diffusion coefficient in a parabolic equation This inverse problem is formulated as a nonlinear operator equation in Hilbert spaces Continuity and differentiability of the corresponding operator is shown In the one-dimensional case uniqueness and conditional stability results are obtained by using the heat equation transform Finally, the problem is solved numerically by iteratively regularized Gauss-Newton method

Approximate inverse for a one-dimensional inverse heat conduction problem

Abstract: In this paper we apply the approximate inverse to a one-dimensional inverse heat conduction problem. We give results about the regularization effect of the approximate inverse and also some error estimate if the solution fulfils some smoothness condition. Then we transfer our theory to an algorithm in pseudo-code which is tested in a numerical example.

Inverse scattering for a locally perturbed half-plane

Abstract: We consider the inverse problem to determine the shape of a local perturbation of a perfectly conducting plate from a knowledge of the far-field pattern of the scattering of TM polarized time-harmonic electromagnetic waves by reformulating it as an inverse scattering problem for a planar domain with corners. For its approximate solution we propose a regularized Newton iteration scheme. For a foundation of Newton type methods we establish the Frechet differentiability of the solution to the scattering problem with respect to the boundary and investigate the injectivity of the linearized mapping. Some numerical examples of the feasibility of the method are presented. For the sake of completeness, the first part of the paper outlines the solution of the direct scattering problem via an integral equation of the first kind including the numerical solution.

Scattering problem for the Zakharov-Shabat equations on the semi-axis

Abstract: This paper is concerned with the direct and inverse scattering problems for compatible differential equations connected with the nonlinear Schrodinger equation (NLSE) on the semi-axis. The corresponding initial boundary value problem (x,t+) was studied recently by Fokas and Its. They found that the key to this problem is to linearize the initial boundary value problem using a Riemann-Hilbert problem. The main goal of this paper is to obtain characteristic properties of the scattering data for compatible differential equations. Our approach uses the transformation operators for both x- and t-equations. For Schwartz type initial and boundary functions we obtain the characteristic properties (A1)-(A5) of the scattering data and derive the so-called xt- and t-integral equations of Marchenko type. The xt-integral equations guarantee the existence of the solution of the NLSE and give an expression of the solution with given scattering data. In turn, the t-integral equations guarantee that one can recover from the scattering data the boundary Dirichlet data v(t) and the corresponding Neumann data w(t) consistent with the given initial function u(x).

Block-iterative interior point optimization methods for image reconstruction from limited data

Abstract: Iterative algorithms for image reconstruction often involve minimizing some cost function h(x) that measures the degree of agreement between the measured data and a theoretical parametrized model In addition, one may wish to have x satisfy certain constraints It is usually the case that the cost function is the sum of simpler functions: h(x) = ∑i = 1Ihi(x) Partitioning the set {i = 1,,I} as the union of the disjoint sets Bn,n = 1,,N, we let hn(x) = ∑iBnhi(x) The method presented here is block iterative, in the sense that at each step only the gradient of a single hn(x) is employed Convergence can be significantly accelerated, compared to that of the single-block (N = 1) method, through the use of appropriately chosen scaling factors The algorithm is an interior point method, in the sense that the images xk + 1 obtained at each step of the iteration satisfy the desired constraints Here the constraints are imposed by having the next iterate xk + 1 satisfy the gradient equation ∇F(xk + 1) = ∇F(xk)-tn∇hn(xk), for appropriate scalars tn, where the convex function F is defined and differentiable only on vectors satisfying the constraints Special cases of the algorithm that apply to tomographic image reconstruction, and permit inclusion of upper and lower bounds on individual pixels, are presented The focus here is on the development of the underlying convergence theory of the algorithm Behaviour of special cases has been considered elsewhere