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Published online by Cambridge University Press: 26 June 2025
This paper is a survey of the inverse scattering problem for time-harmonic acoustic and electromagnetic waves at fixed 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 field data. Included in our discussion is a description of Kirsch's factorization method for solving this problem. We then turn our attention to uniqueness and reconstruction algorithms for determining the support of an inhomogeneous, anisotropic media from acoustic far field data. Our survey is concluded by a brief discussion of the inverse scattering problem for time-harmonic electromagnetic waves.
1. Introduction
The field 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.
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