A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure

TL;DR

This paper introduces a new theoretical and computational approach for 3D inverse medium scattering, utilizing data-driven basis functions based on 3D PSWFs to achieve low-rank approximations and improve imaging accuracy.

Contribution

It develops a novel framework combining 3D prolate spheroidal wave functions with regularization techniques for enhanced inverse scattering analysis.

Findings

Effective low-rank approximation of inverse solutions

Successful imaging of targets despite surrounding noise

Demonstrated potential through numerical examples

Abstract

In this work we propose a theoretical and computational framework for solving the three dimensional inverse medium scattering problem, based on a set of data-driven basis arising from the linearized problem. This set of data-driven basis consists of generalizations of prolate spheroidal wave functions to three dimensions (3D PSWFs), the main ingredients to explore a low-rank approximation of the inverse solution. We first establish the fundamentals of the inverse scattering analysis, including regularity in a customized Sobolev space and new a priori estimate. This is followed by a computational framework showcasing computing the 3D PSWFs and the low-rank approximation of the inverse solution. These results rely heavily on the fact that the 3D PSWFs are eigenfunctions of both a restricted Fourier integral operator and a Sturm-Liouville differential operator. Furthermore we propose a Tikhonov regularization method with a customized penalty norm and a localized imaging technique to image a targeting object despite the possible presence of its surroundings. Finally various numerical examples are provided to demonstrate the potential of the proposed method.