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Regularization and Numerical Solution of the Inverse Scattering Problem using Frames

Kutyniok, Gitta; Mehrmann, Volker; Petersen, Philipp

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Regularization techniques for the numerical solution of nonlinear inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Thikhonov method. We examine both directly the nonlinear problem and a linearized problem obtained by the Born approximation technique. As problem classes we study the acoustic inverse scattering problem and the electromagnetic inverse scattering problem. We show that this approach introduces a sparse regularization for the nonlinear setting and we present a result describing the behavior of the local regularity of a scatterer under linearization, which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by numerical examples for the acoustic inverse scattering problem that highlight the effectiveness of this approach.