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About this book

In the five years since the first edition of this book appeared, the field of in­ verse scattering theory has continued to grow and flourish. Hence, when the opportunity for a second edition presented itself, we were pleased to have the possibility of updating our monograph to take into account recent developments in the area. As in the first edition, we have been motivated by our own view of inverse scattering and have not attempted to include all of the many new directions in the field. However, we feel that this new edition represents a state of the art overview of the basic elements of the mathematical theory of acoustic and electromagnetic inverse scattering. In addition to making minor corrections and additional comments in the text and updating the references, we have added new sections on Newton's method for solving the inverse obstacle problem (Section 5.3), the spectral theory of the far field operator (Section 8.4), a proof of the uniqueness of the solution to the inverse medium problem for acoustic waves (Section 10.2) and a method for determining the support of an inhomogeneous medium from far field data by solving a linear integral equation of the first kind (Section 10.7). We hope that this second edition will attract new readers to the beautiful and intriguing field of inverse scattering.

Table of Contents

Frontmatter

1. Introduction

Abstract
The purpose of this chapter is to provide a survey of our book by placing what we have to say in a historical context. We obviously cannot give a complete account of inverse scattering theory in a book of only a few hundred pages, particularly since before discussing the inverse problem we have to give the rudiments of the theory of the direct problem. Hence, instead of attempting the impossible, we have chosen to present inverse scattering theory from the perspective of our own interests and research program. This inevitably means that certain areas of scattering theory are either ignored or given only cursory attention. In view of this fact, and in fairness to the reader, we have therefore decided to provide a few words at the beginning of our book to tell the reader what we are going to do, as well as what we are not going to do, in the forthcoming chapters.
David Colton, Rainer Kress

2. The Helmholtz Equation

Abstract
Studying an inverse problem always requires a solid knowledge of the theory for the corresponding direct problem. Therefore, the following two chapters of our book are devoted to presenting the foundations of obstacle scattering problems for time-harmonic acoustic waves, that is, to exterior boundary value problems for the scalar Helmholtz equation. Our aim is to develop the analysis for the direct problems to an extent which is needed in the subsequent chapters on inverse problems.
David Colton, Rainer Kress

3. Direct Acoustic Obstacle Scattering

Abstract
This chapter is devoted to the solution of the direct obstacle scattering problem for acoustic waves. As in [41], we choose the method of integral equations for solving the boundary value problems. However, we decided to leave out some of the details in the analysis. In particular, we assume that the reader is familiar with the Riesz-Fredholm theory for operator equations of the second kind in dual systems as described in [41] and [155]. We also do not repeat the technical proofs for the jump relations and regularity properties for single- and double-layer potentials. Leaving aside these two restrictions, however, we will present a rather complete analysis of the forward scattering problem. For the reader interested in a more comprehensive treatment of the direct problem, we suggest consulting our previous book [41] on this subject.
David Colton, Rainer Kress

4. Ill-Posed Problems

Abstract
As previously mentioned, for problems in mathematical physics Hadamard [95] postulated three requirements: a solution should exist, the solution should be unique, and the solution should depend continuously on the data. The third postulate is motivated by the fact that in all applications the data will be measured quantities. Therefore, one wants to make sure that small errors in the data will cause only small errors in the solution. A problem satisfying all three requirements is called well-posed. Otherwise, it is called ill-posed. As shown in the previous chapter, the direct obstacle scattering problem is well-posed.
David Colton, Rainer Kress

5. Inverse Acoustic Obstacle Scattering

Abstract
With the analysis of the preceding chapters, we now are well prepared for studying inverse acoustic obstacle scattering problems. We recall that the direct scattering problem is, given information on the boundary of the scatterer and the nature of the boundary condition, to find the scattered wave and in particular its behavior at large distances from the scatterer, i.e., its far field. The inverse problem starts from this answer to the direct problem, i.e., a knowledge of the far field pattern, and asks for the nature of the scatterer. Of course, there is a large variety of possible inverse problems, for example, if the boundary condition is known, find the shape of the scatterer, or, if the shape is known, find the boundary condition, or, if the shape and the type of the boundary condition are known for a penetrable scatterer, find the space dependent coefficients in the transmission or resistive boundary condition, etc. Here, following the main guideline of our book, we will concentrate on one model problem for which we will develop ideas which in general can also be used to study a wider class of related problems. The inverse problem we consider is, given the far field pattern for one or several incident plane waves and knowing that the scatterer is sound-soft, to determine the shape of the scatterer. We want to discuss this inverse problem for frequencies in the resonance region, that is, for scatterers D and wave numbers k such that the wavelengths 2π/k is less than or of a comparable size to the diameter of the scatterer.
David Colton, Rainer Kress

6. The Maxwell Equations

Abstract
Up until now, we have considered only the direct and inverse obstacle scattering problem for time-harmonic acoustic waves. In the following two chapters, we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves. As in our analysis on acoustic scattering, we begin with an outline of the solution of the direct problem.
David Colton, Rainer Kress

7. Inverse Electromagnetic Obstacle Scattering

Abstract
This last chapter on obstacle scattering is concerned with the extension of the results from Chapter 5 on inverse acoustic scattering to inverse electromagnetic scattering. In order to avoid repeating ourselves, we keep this chapter short by referring back to the corresponding parts of Chapter 5 when appropriate. In particular, for notations and for the motivation of our analysis we urge the reader to get reacquainted with the corresponding analysis in Chapter 5 on acoustics. We again follow the general guideline of our book and consider only one of the many possible inverse electromagnetic obstacle problems: given the electric far field pattern for one or several incident plane electromagnetic waves and knowing that the scattering obstacle is perfectly conducting, find the shape of the scatterer.
David Colton, Rainer Kress

8. Acoustic Waves in an Inhomogeneous Medium

Abstract
Until now, we have only considered the scattering of acoustic and electromagnetic time-harmonic waves in a homogeneous medium in the exterior of an impenetrable obstacle. For the remaining chapters of this book, we shall be considering the scattering of acoustic and electromagnetic waves by an inhomogeneous medium of compact support, and in this chapter we shall consider the direct scattering problem for acoustic waves. We shall content ourselves with the simplest case when the velocity potential has no discontinuities across the boundary of the inhomogeneous medium and shall again use the method of integral equations to investigate the direct scattering problem. However, since boundary conditions are absent, we shall make use of volume potentials instead of surface potentials as in the previous chapters.
David Colton, Rainer Kress

9. Electromagnetic Waves in an Inhomogeneous Medium

Abstract
In the previous chapter, we considered the direct scattering problem for acoustic waves in an inhomogeneous medium. We now consider the case of electromagnetic waves. However, our aim is not to simply prove the electromagnetic analogue of each theorem in Chapter 8, but rather to select the basic ideas of the previous chapter, extend them to the electromagnetic case, and then consider some themes that were not considered in Chapter 8, but ones that are particularly relevant to the case of electromagnetic waves. In particular, we shall consider two simple problems, one in which the electromagnetic field has no discontinuities across the boundary of the medium and the second where the medium is an imperfect conductor such that the electromagnetic field does not penetrate deeply into the body. This last problem is an approximation to the more complicated transmission problem for a piecewise constant medium and leads to what is called the exterior impedance problem for electromagnetic waves.
David Colton, Rainer Kress

10. The Inverse Medium Problem

Abstract
We now turn our attention to the problem of reconstructing the refractive index from a knowledge of the far field pattern of the scattered acoustic or electromagnetic wave. We shall call this problem the inverse medium problem. Of particular interest to us will be the use of a dual space method to determine the refractive index. This method has the numerical advantage of significantly reducing the number of unknowns in the nonlinear optimization step for determining the refractive index and, in addition, leads to a number of mathematical problems which are of interest in their own right. Our aim in this chapter is to develop the theory of the inverse medium problem to the point where an optimization scheme can be formulated for the solution such that under appropriate conditions the infimum of the cost functional is zero. However, since similar optimization schemes were analyzed in depth in Chapters 5 and 7, we shall not dwell on the specific optimization scheme itself, except in Section 10.6 where we present some numerical examples.
David Colton, Rainer Kress

Backmatter

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