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

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