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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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1569-3945
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# Regularization and numerical solution of the inverse scattering problem using shearlet frames

Gitta Kutyniok
/ Volker Mehrmann
/ Philipp C. Petersen
Published Online: 2016-06-29 | DOI: https://doi.org/10.1515/jiip-2015-0048

## Abstract

Regularization techniques for the numerical solution of 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 regularization. We analyze two approaches, namely solvers for the nonlinear problem and for the linearized problem obtained by the Born approximation. As example for the first class we study the acoustic inverse scattering problem, and for the second class, the inverse scattering problem of the Schrödinger equation. Whereas our emphasis for the linearized problem is more on the theoretical side due to the standardness of associated solvers, we provide numerical examples for the nonlinear problem that highlight the effectiveness of our algorithmic approach.

MSC 2010: 34L25; 35P25; 42C40; 42C15; 65J22; 65T60; 76B15; 78A46

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Revised: 2016-03-21

Accepted: 2016-05-04

Published Online: 2016-06-29

Published in Print: 2017-06-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: KU 1446/14

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Research Center Matheon “Mathematics for key technologies”

The first author acknowledges support by the Einstein Foundation Berlin, by the Einstein Center for Mathematics Berlin (ECMath), by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. The second author also acknowledges support by Matheon, and the third author thanks the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” for its support.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 3, Pages 287–309, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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