An inverse Robin problem for Laplace's equation: theoretical results and numerical methods

We consider the problem of detecting corrosion damage on an inaccessible part of a metallic specimen. Electrostatic data are collected on an accessible part of the boundary. The adoption of a simplified model of corrosion appearance reduces our problem to recovering a functional coefficient in a Robin boundary condition for Laplace's equation. We review theoretical results and numerical methods based on the thin-plate approximation and the Galerkin method. Moreover, we introduce a numerical algorithm based on the quasi-reversibility method.