Simultaneous recovery of a corroded boundary and admittance using the Kohn-Vogelius method

TL;DR

This paper presents a new gradient-based method to simultaneously recover an unknown boundary segment and Robin admittance coefficient in a domain using boundary measurements, with proven effectiveness through numerical experiments.

Contribution

It introduces a novel energy-gap based cost function and derives its derivatives to enable joint reconstruction of boundary and admittance, advancing inverse boundary problem techniques.

Findings

Effective numerical reconstruction demonstrated

Method ensures uniqueness with two measurements

Applicable to 1D and 2D domains

Abstract

We address the problem of identifying an unknown portion $\Gamma$ of the boundary of a $d$-dimensional ($d \in \{1, 2\}$) domain $\Omega$ and its associated Robin admittance coefficient, using two sets of boundary Cauchy data $(f, g)$--representing boundary temperature and heat flux--measured on the accessible portion $\Sigma$ of the boundary. Identifiability results \cite{Bacchelli2009,PaganiPierotti2009} indicate that a single measurement on $\Sigma$ is insufficient to uniquely determine both $\Gamma$ and $\alpha$, but two independent inputs yielding distinct solutions ensure the uniqueness of the pair $\Gamma$ and $\alpha$. In this paper, we propose a cost function based on the energy-gap of two auxiliary problems. We derive the variational derivatives of this objective functional with respect to both the Robin boundary $\Gamma$ and the admittance coefficient $\alpha$. These derivatives are utilized to develop a nonlinear gradient-based iterative scheme for the simultaneous numerical reconstruction of $\Gamma$ and $\alpha$. Numerical experiments are presented to demonstrate the effectiveness and practicality of the proposed method.