Exploring low-rank structure for an inverse scattering problem with far-field data

TL;DR

This paper introduces a novel low-rank structure based on prolate spheroidal wave functions for inverse scattering problems, improving stability and robustness in reconstructing unknowns from far-field data.

Contribution

The work develops a new low-rank framework using prolate spheroidal wave functions for inverse scattering, with explicit stability estimates and validated numerical performance.

Findings

Enhanced stability estimates for inverse scattering solutions.

Robustness against noise and modeling errors demonstrated.

Improved resolution capabilities shown through experiments.

Abstract

In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.