Low-rank-assisted inverse medium scattering: Lipschiz stability and ensemble Kalman filter

TL;DR

This paper introduces a low-rank numerical method for inverse medium scattering that achieves Lipschitz stability beyond the Born approximation, utilizing prolate spheroidal wave functions and ensemble Kalman filtering.

Contribution

It develops a low-rank structure based on special eigenfunctions and combines it with an ensemble Kalman filter for improved inverse scattering analysis.

Findings

Lipschitz stability is established for unknowns in a low-rank space.

The method effectively updates unknowns iteratively using ensemble Kalman filtering.

Numerical examples demonstrate the feasibility of the proposed approach.

Abstract

In this work we study the theoretical Lipschitz stability and propose a low-rank-assisted numerical method for the inverse medium scattering beyond the Born region. The proposed low-rank structure is based on the disk prolate spheroidal wave functions, which are eigenfunctions of both the Born forward operator and a Sturm-Liouville differential operator. We obtain Lipschitz stability for unknowns in a low-rank space in the fully nonlinear case and characterize the explicit Lipschitz constant in the linearized region. We further propose an ensemble Kalman filter to iteratively update the unknown in the proposed low-rank space whose dimension is intrinsically determined by the wave number. Moreover the ensembles are sampled according to a novel trace class covariance operator motivated by the connection between the proposed low-rank space and the Sturm-Liouville differential operator. Finally numerical examples are provided to illustrate the feasibility of the proposed method.