Numerical identification of Robin coefficient by a Kohn-Vogelius type regularization method

In this article we consider a Kohn-Vogelius type approach for an inverse Robin problem of an elliptic equation. The unknown Robin coefficient is to be reconstructed with partial boundary measurements, including both Dirichlet and Neumann boundary conditions. Two different boundary value problems are introduced in such a way that two types of boundary measurements are coupled into a single Robin boundary condition, and the two BVPs depend on two different positive numbers alpha(1) and alpha(2), respectively. By applying the Kohn-Vogelius approach with Tikhonov regularization, the boundary data fitting is recast into the whole domain data fitting, which makes the coefficient identification more stable. More importantly, a feasible reconstruction could be obtained even for very small regularization parameter through choosing the values of alpha(1) and alpha(2) properly. Noise model is analysed with data perturbation on both Dirichlet and Neumann boundary data. Theoretical results on stability and solution convergence are also delivered. Numerical examples illustrate the efficiency and stability of the proposed method.