Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

TL;DR

This paper presents a sampling method for reconstructing inclusions with generalized impedance conditions from electrostatic data, demonstrating effectiveness through numerical examples and addressing parameter determination with proven uniqueness.

Contribution

The paper introduces a novel sampling method for inverse shape problems involving generalized impedance conditions, including a proof of uniqueness for material parameter reconstruction.

Findings

Effective reconstruction of inclusions demonstrated numerically.

Uniqueness of material parameter determination proven.

Method applicable to electrostatic inverse problems.

Abstract

In this paper, we derive a Sampling Method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second-order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet-to-Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the material parameters from the Dirichlet-to-Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven.