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The inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves.

In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.

### Chapter 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

### Chapter 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, i.e., 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

### Chapter 3. Direct Acoustic Obstacle Scattering

Abstract
This chapter is devoted to the solution of the direct obstacle scattering problem for acoustic waves.
David Colton, Rainer Kress

### Chapter 4. Ill-Posed Problems

Abstract
As previously mentioned, for problems in mathematical physics Hadamard 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.
David Colton, Rainer Kress

### Chapter 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 2πk is less than or of a comparable size to the diameter of the scatterer. This inverse problem turns out to be nonlinear and improperly posed. Although both of these properties make the inverse problem hard to solve, it is the latter which presents the more challenging difficulties. The inverse obstacle problem is improperly posed since, as we already know, the determination of the scattered wave u s from a given far field pattern u is improperly posed. It is nonlinear since, given the incident wave u i and the scattered wave u s, the problem of finding the boundary of the scatterer as the location of the zeros of the total wave u i + u s is nonlinear.
David Colton, Rainer Kress

### Chapter 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

### Chapter 7. Inverse Electromagnetic Obstacle Scattering

Abstract
This last chapter on obstacle scattering is concerned with the extension of the results from Chap. 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 Chap. 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 Chap. 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

### Chapter 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

### Chapter 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 Chap. 8 but rather to select the basic ideas of the previous chapter, extend them when possible to the electromagnetic case, and then consider some themes that were not considered in Chap. 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

### Chapter 10. Transmission Eigenvalues

Abstract
The transmission eigenvalue problem was previously introduced in Sect. 8.​4 where it was shown to play a central role in establishing the completeness of the set of far field patterns in $$L^2(\mathbb {S}^2)$$. It was then shown in Sect. 8.​6 that the set of transmission eigenvalues was either empty or formed a discrete set, thus leading to the conclusion that except possibly for a discrete set of values of the wave number k > 0, the set of far field patterns is complete in $$L^2(\mathbb {S}^2)$$. In this chapter we return to the subject of transmission eigenvalues and consider further topics of interest. In particular, we begin by showing the existence of transmission eigenvalues and then deriving a monotonicity result for the first positive transmission eigenvalue. We then proceed to describe a boundary integral equation approach to the transmission eigenvalue problem, the existence of complex transmission eigenvalues in the case of a spherically stratified medium, and the inverse spectral problem for the case of such a medium. We conclude this chapter by considering a modified transmission eigenvalue problem in which the wave number k > 0 is kept fixed and the eigenparameter is now an artificial coefficient introduced through the use of a modified far field operator. Our analysis is restricted to the case of acoustic waves.
David Colton, Rainer Kress

### Chapter 11. 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. We first consider the case of acoustic waves and the use of the Lippmann–Schwinger equation to reformulate the acoustic inverse medium problem as a problem in constrained optimization. Included here is a brief discussion of the use of the Born approximation to linearize the problem. We then proceed to the proof of a uniqueness theorem for the acoustic inverse medium problem. Our uniqueness result is then followed by a discussion of decomposition methods for solving the inverse medium problem for acoustic waves and the use of sampling methods and transmission eigenvalues to obtain qualitative estimates on the refractive index. We conclude by examining the use of decomposition methods to solve the inverse medium problem for electromagnetic waves followed by some numerical examples illustrating the use of decomposition methods to solve the inverse medium problem for acoustic waves.
David Colton, Rainer Kress