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Point sources and multipoles in inverse scattering theory. (English) Zbl 0985.78016
Chapman & Hall/CRC Research Notes in Mathematics. 427. Boca Raton, FL: Chapman & Hall/CRC. xi, 261 p. (2001).
The book provides a very readable survey of recent developments in inverse acoustic and electromagnetic scattering, focusing on methods developed over the last years by Colton, Kirsch, and the author. It can be recommended to everyone interested in the analysis and numerics of scattering by obstacles and media.
Chapter 1 presents a general introduction to reconstruction methods, and the basic definitions and tools. Chapter 2 is devoted to the analysis of direct scattering problems and contains, in particular, estimates for the scattered field of point-sources and multipoles. Uniqueness and stability results for the reconstruction of the shape of a scatterer from the knowledge of the full far field patterns are given in Chapter 3. The practically important finite data case is investigated in Chapter 4, based on the author’s concept of $$\varepsilon$$-uniqueness and $$\varepsilon$$-stability.
The last three chapters are dedicated to the analysis and performance of reconstruction methods for acoustic and electromagnetic obstacle scattering. Chapter 5 is devoted to a point-source method for impenetrable scatterers. Chapter 6 proposes the method of singular sources for recovering both impenetrable and penetrable scatterers, which uses the singular behaviour of scattered fields of multipoles. Chapter 7 introduces linear sampling methods and includes the reconstruction of inhomogeneous isotropic, orthotropic and anisotropic media.

##### MSC:
 78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory 78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory 35R30 Inverse problems for PDEs 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 65R32 Numerical methods for inverse problems for integral equations 45Q05 Inverse problems for integral equations