Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data

TL;DR

This paper extends linear sampling and factorization methods to recover buried corrosion boundaries from partial electrostatic measurements, providing new theoretical and numerical tools for non-destructive testing of hidden objects.

Contribution

It introduces novel extensions of shape and parameter recovery methods to electrostatic problems with partial data, including boundary integral equations for Green's function recovery.

Findings

Successfully applied extended methods to inverse shape problems.

Proved uniqueness and stability for inverse parameter problems.

Developed boundary integral equations for Green's function reconstruction.

Abstract

We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the non-destructive testing for a partially buried object from electrostatic measurements on the accessible part of the boundary. The main novelty is the extension of the linear sampling and factorization methods to an electrostatic problem with partial measurements. These methods so far have only mainly applied to recovering interior defects which is a simpler problem. Another important aspect of this paper is in our numerics, where we derive a system of boundary integral equations to recover the mixed Green's function which is needed for our inversion. With this, we are able to analytically and numerically solve the inverse shape problem. For the inverse parameter problem, we prove uniqueness and Lipschitz-stability (in a finite dimensional function space) assuming that one has the associated Neumann-to-Dirichlet operator on the accessible part of the boundary.