Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames

TL;DR

This paper introduces shearlet-based regularization techniques for solving inverse scattering problems, demonstrating their effectiveness through analytical results and numerical experiments for both linearized and nonlinear cases.

Contribution

It proposes a novel shearlet frame regularization approach for inverse scattering problems, analyzing both nonlinear and linearized solutions with theoretical and numerical validation.

Findings

Shearlet frames effectively sparsify boundary features of scatterers.

The proposed methods improve solution accuracy in inverse scattering.

Numerical results confirm the approach's effectiveness for nonlinear problems.

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 in a Tikhonov method. We analyze two approaches, namely solvers for the nonlinear problem and for the linearized problem obtained by the Born approximation technique. As example for the first class we study the acoustic inverse scattering problem, and for the second class, the inverse scattering problem of the Schr\"{o}dinger equation. In both cases, we derive analytical results for our approaches. 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.