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

TL;DR

This paper addresses the simultaneous recovery of an unknown boundary Γ and the Robin coefficient α in a domain Ω from two Cauchy data pairs on Σ. It adopts a Kohn–Vogelius energy-gap functional $J(ω, α)$ and derives its shape derivative with respect to ω and the Fréchet derivative with respect to α, enabling a gradient-based, joint reconstruction. The method uses a Sobolev gradient to produce smooth descent directions and solves the state equations via FEM, avoiding adjoint equations for the gradient. Numerical experiments on circular exterior domains show reliable recovery of Γ and qualitative recovery of α under exact data; with noisy data, multiple measurements are needed for stable reconstruction, highlighting potential avenues for future work.

Abstract

We address the problem of identifying an unknown portion $Γ$ of the boundary of a $d$-dimensional ($d \in \{1, 2\}$) domain $Ω$ 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 $Σ$ of the boundary. Identifiability results \cite{Bacchelli2009,PaganiPierotti2009} indicate that a single measurement on $Σ$ is insufficient to uniquely determine both $Γ$ and $α$, but two independent inputs yielding distinct solutions ensure the uniqueness of the pair $Γ$ and $α$. 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 $Γ$ and the admittance coefficient $α$. These derivatives are utilized to develop a nonlinear gradient-based iterative scheme for the simultaneous numerical reconstruction of $Γ$ and $α$. Numerical experiments are presented to demonstrate the effectiveness and practicality of the proposed method.