Least-Squares Method for Inverse Medium Problems

TL;DR

This paper introduces a two-stage least-squares approach for inverse medium problems, combining direct sampling for localization and total least-squares with regularization for accurate reconstruction, demonstrating robustness to noisy data.

Contribution

The paper proposes a novel two-stage least-squares method with mixed regularization for improved accuracy in reconstructing medium properties from noisy data.

Findings

Effective localization of inhomogeneities using direct sampling.

Enhanced reconstruction accuracy with total least-squares and regularization.

Robust performance demonstrated under noisy conditions.

Abstract

We present a two-stage least-squares method to inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with great tolerance of noise.