Another in the acclaimed series by the Society for Industrial and Applied Mathematics (SIAM) in collaboration with the National Science Foundation, this book is dedicated to the analytical and computational aspects of inverse problems, an area with a rich history of mathematical development.
Specifically, it centers on understanding the behavior of solutions of the electromagnetic scattering problem, &dgr;u + k2 n(x) u = 0 in R2 R3 subject to boundary conditions at infinity, where n is the index of refraction, and k is the wave number.
The index of refraction generally suffers a discontinuity in the presence of a scatterer, and the inverse problem is to determine the location and the nature of the scatterer from the far-field knowledge of u. As is the case with nearly all inverse problems of interest, this boundary value problem is highly nonlinear and ill-posed, thus leading to a remarkably challenging computational problem, especially in R3.
The book begins with an excellent introduction to this problem, describing the variety of approaches taken to obtain reliable and robust approximate solutions to the inverse scattering problem. The authors describe Born approximation, regularization methods, and the iterative and optimization approaches, before settling on the linear sampling method (LSM), the main topic of this book. They describe the strengths and weaknesses of each approach, including LSM, in detail.
One of the refreshing aspects of the first chapter is the authors’ willingness to not only list the weaknesses of LSM, but to point out to the reader that LSM is somewhat controversial; a few relevant references are listed for the interested reader to pursue. Nevertheless, the authors convince the reader that the strengths of this method outweigh its weaknesses. They dedicate the remaining chapters to the development of the theoretical aspects of LSM, its generalizations, and their computational implementations.
This book is an excellent companion to the 1998 text [1] by Colton and Kress. It would benefit any science graduate student who wishes to know the state of the art in this field. The authors present the material well, and make it accessible to any reader with a strong background in mathematical analysis.
