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Preprint 26-2014

Regularization and Numerical Solution of the Inverse Scattering Problem using Frames

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Author(s) : Gitta Kutyniok , Volker Mehrmann , Philipp Petersen

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 26-2014

MSC 2000

35P25 Scattering theory for PDE
65T60 Wavelets

Abstract :
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.

Keywords : Helmholtz equation, Inverse medium scattering, Regularization, Schrödinger equation, Shearlets, Sparse approximation

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