This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. The second edition includes material on Newton s method for the inverse obstacle problem, an elegant proof of uniqueness for the inverse medium problem, a discussion of the spectral theory of the far field operator and a method for determining the support of an inhomogeneous medium from far field data.
Focusing on qualitative methods in inverse scattering theory, this book introduces innovative approaches that bypass traditional weak scattering assumptions and nonlinear optimization techniques. It emphasizes sampling methods and transmission eigenvalues to estimate the shape and material properties of scattering objects with minimal prior information. To accommodate readers from non-mathematical backgrounds, it includes essential concepts from functional analysis, Sobolev spaces, ill-posed problems, and complex variable theory. This edition builds upon the authors' earlier work, enhancing its accessibility and depth.
Presenting topics that have not previously been contained in a single volume, this book offers an up-to-date review of computational methods in electromagnetism, with a focus on recent results in the numerical simulation of real-life electromagnetic problems and on theoretical results that are useful in devising and analyzing approximation algorithms. Based on four courses delivered in Cetraro in June 2014, the material covered includes the spatial discretization of Maxwell’s equations in a bounded domain, the numerical approximation of the eddy current model in harmonic regime, the time domain integral equation method (with an emphasis on the electric-field integral equation) and an overview of qualitative methods for inverse electromagnetic scattering problems. Assuming some knowledge of the variational formulation of PDEs and of finite element/boundary element methods, the book is suitable for PhD students and researchers interested in numerical approximation of partial differential equations and scientific computing.