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

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.