Regularized reduced order Lippmann-Schwinger-Lanczos method for inverse scattering problems in the frequency domain

Abstract

Inverse scattering is broadly applicable in quantum mechanics, remote sensing, geophysical, and medical imaging. This paper presents a robust direct non-iterative reduced order model (ROM) method for solving inverse scattering problems based on an efficient approximation of the resolvent operator, resulting in the regularized Lippmann-Schwinger-Lanczos (LSL) algorithm. We show that the efficiency of the method relies upon the weak dependence of the orthogonalized basis on the unknown potential in the Schrödinger equation by demonstrating that the Lanczos orthogonalization is equivalent to performing Gram-Schmidt on the ROM time snapshots. We then develop the LSL algorithm in the frequency domain with two levels of regularization. The proposed bi-level regularization of the algorithm represents a significant advancement in computational stability, enabling its application to real data sets that are larger than used previously with LSL and inherently contain errors. We show that the same procedure can be extended beyond the Schrödinger formulation to the diffusive Helmholtz equation, e.g., to imaging the conductivity using diffusive electromagnetic fields in conductive media with localized positive conductivity perturbations. Numerical experiments for diffusive Helmholtz and Schrödinger problems show that the proposed bi-level regularization scheme significantly improves the performance of the LSL algorithm, allowing for accurate reconstructions with noisy data and large data sets.