A Stable Iterative Direct Sampling Method for Elliptic Inverse Problems with Partial Cauchy Data

TL;DR

This paper introduces a robust iterative direct sampling method for elliptic inverse problems with partial data, improving stability, resolution, and noise resistance in reconstructing inhomogeneities across various applications.

Contribution

It presents a novel IDSM combining data completion, heterogeneously regularized DtN map, and stabilization strategies, advancing the accuracy and stability of inverse problem solutions.

Findings

Enhanced reconstruction accuracy in numerical experiments

Robustness against measurement noise demonstrated

Flexible measurement configuration applicability

Abstract

We develop a novel iterative direct sampling method (IDSM) for solving linear or nonlinear elliptic inverse problems with partial Cauchy data. It integrates three innovations: a data completion scheme to reconstruct missing boundary information, a heterogeneously regularized Dirichlet-to-Neumann map to enhance the near-orthogonality of probing functions, and a stabilization-correction strategy to ensure the numerical stability. The resulting method is remarkably robust with respect to measurement noise, is flexible with the measurement configuration, enjoys provable stability guarantee, and achieves enhanced resolution for recovering inhomogeneities. Numerical experiments in electrical impedance tomography, diffuse optical tomography, and cardiac electrophysiology show its effectiveness in accurately reconstructing the locations and geometries of inhomogeneities.