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Qualitative methods in inverse scattering theory. An introduction. (English) Zbl 1099.78008
Interaction of Mechanics and Mathematics. Berlin: Springer (ISBN 3-540-28844-9/pbk). viii, 227 p. (2006).
This book presents a qualitative approach to inverse scattering theory for electromagnetic waves, and contains also complementary material for the corresponding direct problems. It consists of 9 chapters, references and an index.
The topics are presented at an introductional level, accessible to anyone having a mathematical background only in advanced calculus and linear algebra. The necessary basic concepts on functional analysis, Sobolev spaces and ill-posed problems are provided in the first two chapters.
The third chapter treats the problem of scattering of a time harmonic plane wave by an imperfectly conducting infinite cylinder. It includes an introduction to the Maxwell equations, the Sommerfeld radiation condition, and to the relevant properties of Bessel and Hankel functions. The Vekua-Rellich lemma is also deduced, and the potential method is applied to the problem.
Chapter 4 introduces the inverse scattering problem for an impedance conductor in view of determining the support of the scattering object and the surface impedance (when given the far field pattern of the scattered field). The used approach is based on the linear sampling method (proposed in [D. Colton and A. Kirsch, Inverse Probl. 12, No. 4, 383–393 (1996; Zbl 0859.35133)] and [D. Colton, M. Piana and R. Potthast, Inverse Probl. 13, No. 6, 1477–1493 (1997; Zbl 0902.35123)]).
In Chapter 5 it is performed the analysis of the solution to the direct problem of scattering of incident time harmonic electromagnetic waves by a penetrable orthotropic infinite cylinder. The used model consists of a transmission problem for the Helmholtz equation outside the scatterer and an equation with nonconstant coefficients inside the scatterer. In here, the approach is based on a variational method in a framework of Bessel potential spaces.
As a natural sequence of the previous chapter, in Chapter 6 the inverse scattering problem for an orthotropic medium is studied. The goal is to determine the support of the orthotropic inhomogeneity given the far field pattern of the scattered field for many incident directions.
Chapter 7 is mainly devoted to the exemplification of some of the consequences of factorizing (normal) far field operators originated by some scattering problems. In this sense, the factorization method introduced by A. Kirsch is here applied to reconstruct the shape of a perfect conductor from the knowledge of the far field operator.
In Chapter 8 it is presented the mathematical analysis of two mixed boundary value problems generated by (i) the scattering by a perfect conductor that is partially coated by a thin dielectric layer and (ii) the scattering by an orthotropic dielectric that is partially coated by a thin layer of highly conducting material. Additionally, in the last two sections the analysis of some direct and inverse problems of scattering by cracks is included. Namely, for the direct problem it is used the model situation of the scattering of a time harmonic electromagnetic plane wave by an infinite cylinder having an open arc as cross section.
The last chapter, “A glimpse at Maxwell’s equations”, presents some details for treating three dimensional electromagnetic scattering problems, and may be viewed as a complement to the other chapters where the used models allow the reduction of the Maxwell system to a two dimensional scalar equation.
The book is well-written, and readable at the level of advanced graduate students.

MSC:
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35R30 Inverse problems for PDEs
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
78-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory
35J25 Boundary value problems for second-order elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
47A40 Scattering theory of linear operators
35P25 Scattering theory for PDEs
47H50 Potential operators (MSC2000)
78A45 Diffraction, scattering
31A10 Integral representations, integral operators, integral equations methods in two dimensions
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