Inverse scattering theory is an important area of applied mathematics
due to its central role in such areas as medical imaging, nondestructive
testing and geophysical exploration. Until recently all existing
algorithms for solving inverse scattering problems were based on using
either a weak scattering assumption or on the use of nonlinear
optimization techniques. The limitations of these methods have led in
recent years to an alternative approach to the inverse scattering
problem which avoids the incorrect model assumptions inherent in the use
of weak scattering approximations as well as the strong a priori
information needed in order to implement nonlinear optimization
techniques. These new methods come under the general title of
qualitative methods in inverse scattering theory and seek to determine
an approximation to the shape of the scattering object as well as
estimates on its material properties without making any weak scattering
assumption and using essentially no a priori information on the nature
of the scattering object. This book is designed to be an introduction to
this new approach in inverse scattering theory focusing on the use of
sampling methods and transmission eigenvalues. In order to aid the
reader coming from a discipline outside of mathematics we have included
background material on functional analysis, Sobolev spaces, the theory
of ill posed problems and certain topics in in the theory of entire
functions of a complex variable. This book is an updated and expanded
version of an earlier book by the authors published by Springer titled
Qualitative Methods in Inverse Scattering Theory
Review of Qualitative Methods in Inverse Scattering Theory All in all,
the authors do exceptionally well in combining such a wide variety of
mathematical material and in presenting it in a well-organized and
easy-to-follow fashion. This text certainly complements the growing body
of work in inverse scattering and should well suit both new researchers
to the field as well as those who could benefit from such a nice
codified collection of profitable results combined in one bound volume.
SIAM Review, 2006
