Recursive linearization method for inverse medium scattering problems with complex mixture Gaussian error learning

TL;DR

This paper introduces a new iterative method called GMRLM for inverse medium scattering problems that models and learns complex Gaussian mixture errors to improve stability and accuracy in numerical solutions.

Contribution

It extends the recursive linearization method by incorporating complex Gaussian mixture error modeling and learning, enhancing the robustness of inverse scattering solutions.

Findings

GMRLM effectively accounts for modeling errors in IMSP.

The method improves stability and accuracy over traditional approaches.

Numerical examples demonstrate its practical effectiveness.

Abstract

This paper is concerned with the modeling errors appeared in the numerical methods of inverse medium scattering problems (IMSP). Optimization based iterative methods are wildly employed to solve IMSP, which are computationally intensive due to a series of Helmholtz equations need to be solved numerically. Hence, rough approximations of Helmholtz equations can significantly speed up the iterative procedure. However, rough approximations will lead to instability and inaccurate estimations. Using the Bayesian inverse methods, we incorporate the modelling errors brought by the rough approximations. Modelling errors are assumed to be some complex Gaussian mixture (CGM) random variables, and in addition, well-posedness of IMSP in the statistical sense has been established by extending the general theory to involve CGM noise. Then, we generalize the real valued expectation-maximization (EM) algorithm used in the machine learning community to our complex valued case to learn parameters in the CGM distribution. Based on these preparations, we generalize the recursive linearization method (RLM) to a new iterative method named as Gaussian mixture recursive linearization method (GMRLM) which takes modelling errors into account. Finally, we provide two numerical examples to illustrate the effectiveness of the proposed method.