Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems

TL;DR

This paper introduces a novel direct nonlinear inversion method combining data-driven reduced order models with the Lippmann-Schwinger equation, significantly improving inverse scattering accuracy without iterative updates.

Contribution

The paper presents a new approach that integrates ROMs with the Lippmann-Schwinger equation, enabling efficient and accurate direct inversion in scattering problems.

Findings

Inversion outperforms Born inversion in spectral domain data.

Method achieves accuracy comparable to known internal solutions.

ROM sparsity enhances computational efficiency.

Abstract

Data-driven reduced order models (ROMs) are combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.