Reconstruction of material losses by perimeter penalization and phase-field methods

TL;DR

This paper presents a numerical variational method using phase-field functions and perimeter penalization to reconstruct material losses like cavities in conducting bodies from boundary electrostatic measurements, with proven convergence.

Contribution

It introduces a novel phase-field based variational approach with perimeter penalization for inverse material loss problems, including a convergence proof.

Findings

Successful reconstruction of cavities from boundary data

Convergence of the method as measurement error decreases

Effective regularization of the ill-posed inverse problem

Abstract

We treat the inverse problem of determining material losses, such as cavities, in a conducting body, by performing electrostatic measurements at the boundary. We develop a numerical approach, based on variational methods, to reconstruct the unknown material loss by a single boundary measurement of current and voltage type. The method is based on the use of phase-field functions to model the material losses and on a perimeter-like penalization to regularize the otherwise ill-posed problem.We justify the proposed approach by a convergence result, as the error on the measurement goes to zero.