Surveys on Solution Methods for Inverse Problems

It has only been since the mid-1960s that inverse problems has been identified as a proper subfield of mathematics. Prior to this conventional wisdom held it was not an area appropriate for mathematical analysis. This historical prejudice dates back to Hadamard who claimed that the only problems of physical interest were those that had a unique solution depending continuously on the given data. Such problems were well-posed and problems that were not well-posed were labeled ill-posed. In particular, ill-posed problems connected with partial differential equations of mathematical physics were considered to be of purely academic interest and not worthy of serious study. In the meantime, the success of radar and sonar during the Second World War caused scientists to ask the question if more could be determined about a scattering object than simply its location. Such problems are in the category of inverse scattering problems and it was slowly realised that these problems, although of obvious physical interest, were ill-posed mathematically. Similar problems began to present themselves in other areas such as geophysics, medical imaging and non-destructive testing. However, due to the lack of a mathematical theory of inverse problems together with limited computational capabilities, further progress was not possible.

The growth of the area of inverse problems within applied mathematics in recent years has been driven both by the needs of applications and by advances in a rigorous convergence theory of regularization methods for the solution of nonlinear ill-posed problems. There are at least two widely used approaches for solving inverse problems in a stable way: Tikhonov regularization and iterative regularization techniques. In this paper we give an overview over the latter. Moreover, we put the analysis of iterative methods for the solution of ill-posed problems into perspective with the analysis of iterative methods for the solution of well-posed problems.

In this survey we review recent developments concerning the efficient iterative regularization of image reconstruction problems in atmospheric imaging. We present a number of preconditioners for the minimization of the corresponding Tikhonov functional, and discuss the alternative of terminating the iteration early, rather than adding a stabilizing term in the Tikhonov functional. The methods are examplified for a (synthetic) model problem.

We survey continuous and discrete regularization methods for first-kind Volterra problems with continuous kernels. Classical regularization methods tend to destroy the non-anticipatory (or causal) nature of the original Volterra problem because such methods typically rely on computation of the Volterra adjoint operator, an anticipatory operator. In this survey we pay special attention to particular regularization methods, both classical and nontraditional, which tend to retain the Volterra structure of the original problem. Our attention will primarily be focused on linear problems, although extensions of methods to nonlinear and integro-operator Volterra equations are mentioned when known.

We describe a rigorous layer stripping approach to inverse scattering for the Helmholtz equation in one dimension. In section 3, we show how the Ricatti ordinary differential equation, which comes from the invariant embedding approach to forward scattering, becomes an inverse scattering algorithm when combined with the principle of causality. In section 4 we discuss a method of stacking and splitting layers. We first discuss a formula for combining the reflection coefficients of two layers to produce the reflection coefficient for the thicker layer built by stacking the first layer upon the second. We then describe an algorithm for inverting this procedure; that is, for splitting a reflection coefficient into two thinner reflection coefficients. We produce a strictly convex variational problem whose solution accomplishes this splitting. Once we can split an arbitrary layer into two thinner layers, we proceed recursively until each reflection coefficients in the stack is so thin that the Born approximation holds (i.e. the reflection coefficient is approximately the Fourier transform of the derivative of the logarithm of the wave speed). We then invert the Born approximation in each thin layer.

A survey is given of the linear sampling method for solving the inverse scattering problem of determing the support of an inhomogeneous medium from a knowledge of the far field pattern of the scattered field. An application is given to the problem of detecting leukemia in the human body.

Carleman estimates are a powerful tool which was originally proposed by T. Carleman in 1939 for proofs of uniqueness results for ill-posed Cauchy problems. Since 1981 this tool has been applied to inverse problems for PDEs. The goal of this paper is to provide a tutorial-like short review of the role which Carleman estimates play in three fundamental issues of inverse problems: uniqueness, stability, and numerical methods.

Tomographic methods are described that will reconstruct object boundaries in shallow water using sonar data. The basic ideas involve microlocal analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model.

The aim of this paper is to describe and analyse functionals which can be used for computing hyperthermia treatment plans. All these functionals have in common that they can be optimised by efficient numerical methods. These methods have been implemented and tested with real data from the Rudolf Virchow Klinikum, Berlin. The results obtained by these methods are comparable to those obtained by comparatively expansive global optimisation techniques.

We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indirect and nonlinear. In this paper we describe three distinct approaches to this problem. In the first approach the data is mode shape level sets and frequencies. Here formulas for approximate solutions are given based on perturbation results. In the second approach the data is frequencies and boundary mode shape measurements; uniqueness results are obtained using the boundary control method. In the third approach the data is frequencies for four boundary value problems. Local existence, uniqueness results are established together with numerical results for approximate solutions.

Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media.

We consider the inverse problem of determining a Riemannian metric in R ⁿ which is euclidean outside a ball from scattering information. This is a basic inverse scattering problem in anisotropic media. By looking at the wave front set of the scattering operator we are led to consider the “classical” problem of determining a Riemannian metric by measuring the travel times of geodesics passing through the domain. We survey some recent developments on this problem.

What mathematicians, scientists, engineers, and statisticians mean by “inverse problem” differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results.