Inverse Scattering Theory and Transmission Eigenvalues

Inverse scattering theory is a major theme of applied mathematics, and it has applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting particular problems in the development of efficient inversion algorithms. Although linearized models continue to play an important role in many applications, an increased need to focus on problems in which multiple scattering effects cannot be ignored has led to a central role for nonlinearity, and the possibility of collecting large amounts of data over limited regions of space means that the ill-posed nature of the inverse scattering problem has become a problem of central importance.

Initial efforts to address the nonlinear and the ill-posed nature of the inverse scattering problem focused on nonlinear optimization methods. While efficient in many situations, strong a priori information is necessary for their implementation. This problem led to a qualitative approach to inverse scattering theory in which the amount of a priori information is drastically reduced, although at the expense of only obtaining limited information about the values of the constitutive parameters. This qualitative approach (the linear sampling method, the factorization method, the theory of transmission eigenvalues, etc.) is the theme of Inverse Scattering Theory and Transmission Eigenvalues.

The authors begin with a basic introduction to the theory, then proceed to more recent developments, including a detailed discussion of the transmission eigenvalue problem; present the new generalized linear sampling method in addition to the well-known linear sampling and factorization methods; and in order to achieve clarification of presentation, focus on the inverse scattering problem for scalar homogeneous media.

Audience: This book is for research mathematicians as well as engineers and physicists working on problems in target identification. It will also be of interest to advanced graduate students in diverse areas of applied mathematics.

Contents: Preface; Chapter 1: Inverse Scattering Theory; Chapter 2: The Determination of the Support of Inhomogeneous Media; Chapter 3: The Interior Transmission Problem; Chapter 4: The Existence of Transmission Eigenvalues; Chapter 5: Inverse Spectral Problems for Transmission Eigenvalues; Bibliography; Index

Fioralba Cakoni is a professor in the Department of Mathematics at Rutgers University. She is coauthor with David Colton of A Qualitative Approach to Inverse Scattering Theory (Springer, 2014).

David Colton is a professor in the Department of Mathematical Sciences at the University of Delaware, where he was appointed Unidel Professor in 1996. He is coauthor of the aforementioned book with Fioralba Cakoni and coauthor of Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 3rd edition, 2013) with Rainer Kress.

Houssem Haddar is Director of Research at INRIA and a part-time Professor at Ecole Polytechnique. He is coauthor with Ralf Hiptmair, Peter Monk, and Rodolfo Rodriguez of Computational Electromagnetism (Springer, 2015).