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Ever since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hülsmeyer devised a way to determine the distance to the object. At that rate of improvement one might have extrapolated to unimaginable twenty-first-century capabilities. Unfortunately, what has proved to be unimaginable is the difficulty of doing much more than the original device had already accomplished. It would seem that some forms of bionic vision have gone the way of rocket backpacks -that is, until recently.The newest book by Andreas Kirsch with coauthor Natalia Grinberg, The Factorization Method for Inverse Problems, collects over a decade of work by Kirsch and collaborators on a simple method for shape identification in inverse scattering. This book belongs to the next generation of monographs on inverse scattering following the now standard works of Colton and Kress [2] (Inverse Acoustic and Electromagnetic Scattering Theory (1998)) and Isakov [7] (Inverse Problems for Partial Differential Equations).Kirsch's factorization method arose from experimentation with noniterative inverse scattering methods that avoid the computational expense of calculating the solution to the forward problem at each iteration. Noniterative methods attack head-on the inverse problem of determining the scatterer from measured scattered fields by attempting, in principle, simply to invert the scattering operator. In most situations of interest, however, the scattering operator is nonlinear and the inverse problem is ill-posed. Early ideas focused on operator splitting techniques that decompose the scattering operator into a well-posed nonlinear part and an ill-posed linear part, each of which can be inverted stably. Another class of noniterative methods use indicator functions to detect the inconsistency or unsolvability of an easily computed auxiliary problem parameterized by points in space. The shape and location of the object is then determined by those points where the auxiliary problem is solvable. This latter generation of techniques, of which the factorization method is one, is both stable and computationally fast. What separates the factorization method from most of the other noniterative techniques is that it is mathematically complete: the computable criterion for determining the shape and location of the scatterers is both sufficient and necessary while most other techniques rely on only sufficient criteria.The book consists of seven chapters treating the application of the factorization method for, respectively: simplest cases (namely where the far field operator is normal), refinements for more complicated settings (namely, where the far field operator is not normal), so-called mixed boundary value scattering problems, scattering from inhomogeneous media, Maxwell's equations, impedance tomography, and finally a short ...

This paper is concerned with the inverse obstacle scattering problem for time harmonic plane waves. We derive a factorization of the far field operator F in the form GSG * and prove that the ranges of √ |F | and G coincide. Then we give an explicit characterization of the scattering obstacle which uses only the spectral data of the far field operator F . This result is used to prove a convergence result for a recent numerical method proposed by Colton, Kirsch, Monk, Piana and Potthast. We illustrate this method by some numerical examples.

This paper is concerned with the development of an inversion scheme for twodimensional inverse scattering problems in the resonance region which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer. It is assumed that the far field pattern corresponding to observation angle and plane waves incident at angle is known for all. From this information, the support of the scattering obstacle is obtained by solving the integral equation where k is the wavenumber and is on a rectangular grid containing the scatterer. The support is found by noting that is unbounded as approaches the boundary of the scattering object from inside the scatterer. Numerical examples are given showing the practicality of this method.

), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface of the First EditionFollowing Keller [136] we call two problems inverse to each other if the formulation of each of them requires full or partial knowledge of the other. By this definition, it is obviously arbitrary which of the two problems we call the direct and which we call the inverse problem. But usually, one of the problems has been studied earlier and, perhaps, in more detail. This one is usually called the direct problem, whereas the other is the inverse problem. However, there is often another more important difference between these two problems. Hadamard (see [103]) introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem ill-posed. It turns out that many interesting and important inverse problems in science lead to ill-posed problems, whereas the corresponding direct problems are well-posed. Often, existence and uniqueness can be forced by enlarging or reducing the solution space (the space of "models"). For restoring stability, however, one has to change the topology of the spaces, which is in many cases impossible because of the presence of measurement errors. At first glance, it seems to be impossible to compute the solution of a problem numerically if the solution of the problem does not depend continuously on the data, that is, for the case of ill-posed problems. Under additional a priori information about the solution, such as smoothness and bounds on the derivatives, however, it is possible to restore stability and construct efficient numerical algorithms.We make no claim to cover all of the topics in the theory of inverse problems. Indeed, with the rapid growth of this field and its relationship to many fields of natural and technical sciences, such a task would certainly be impossible for a single author in a single volume. The aim of this book is twofold: first, we introduce the reader to the basic notions and difficulties encountered with ill-posed problems. We then study the basic properties of regularization methods for linear ill-posed problems. These methods can roughly be classified into two groups, namely, whether the regularization parameter is chosen a priori or a posteriori. We study some of the most important regularization schemes in detail.The second a...