Inverse Acoustic and Electromagnetic Scattering Theory: Inverse Acoustic and Electromagnetic Scattering Theory.pdf

Inverse Acoustic and Electromagnetic
Scattering Theory
DAVID COLTON
Abstract. This paper is a survey of the inverse scattering problem for
time-harmonic acoustic and electromagnetic waves at xed frequency. We
begin by a discussion of weak scattering and Newton-type methods for
solving the inverse scattering problem for acoustic waves, including a brief
discussion of Tikhonov's method for the numerical solution of ill-posed
problems. We then proceed to prove a uniqueness theorem for the inverse
obstacle problems for acoustic waves and the linear sampling method for
reconstructing the shape of a scattering obstacle from far eld data. In-
cluded in our discussion is a description of Kirsch's factorization method
for solving this problem. We then turn our attention to uniqueness and re-
construction algorithms for determining the support of an inhomogeneous,
anisotropic media from acoustic far eld data. Our survey is concluded
by a brief discussion of the inverse scattering problem for time-harmonic
electromagnetic waves.
1. Introduction
The eld of inverse scattering, at least for acoustic and electromagnetic waves,
can be viewed as originating with the invention of radar and sonar during the
Second World War. Indeed, as every viewer of World War II movies knows, the
ability to use acoustic and electromagnetic waves to determine the location of
hostile objects through sea water and clouds played a decisive role in the outcome
of that war. Inspired by the success of radar and sonar, the prospect was raised
of the possibility of not only determining the range of an object from the transmitter,
but to also image the object and thereby identify it, i.e. to distinguish
between a whale and a submarine or a goose and an airplane. However, it was
soon realized that the problem of identication was considerably more dicult
than that of simply determining the location of a target. In particular, not only
was the identication problem computationally extremely expensive, and indeed
beyond the capabilities of post-war computing facilities, but the problem was
also ill-posed in the sense that the solution did not depend continuously on the

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