A Robust Recovery Algorithm for the Robin Inverse Problem

We consider the inverse problem of identifying a Robin coefficient by performing measurements on some part of the boundary. By using a Kohn and Vogelius cost function, the inverse problem is turned into an optimisation one, which is proved to be stable with respect to the data. Robustness of the so obtained algorithm is also proved. Further investigations are related the recovery of the fast variations of the Robin coefficient, which are relevant to locate the corrosion and evaluate its level. 1 The Robin inverse problem Corrosion of materials impacts the impedance coefficient, which intervines in a Robin boundary condition, and identifying this coefficient may thus be a way to locate the corrosion, and possibly evaluate its level, in some structure by an electric impedance tomography process. The forward problem is modelled by the Laplace equation, with a Robin boundary condition on the possibly corroded part of the boundary, whereas we hold two boundary conditions on the remaining part of the boundary : a Neumann one standing for the imposed current flux, and a Dirichlet one standing for the measured potential. Let Ω be a connected bounded domain of R. The boundary ∂Ω is assumed to be a C Jordan curve, for some β ∈]0, 1[. Morever, let γ and ΓN be two non empty connected open subsets of ∂Ω such that: ∂Ω = γ ∪ ΓN The inverse problem (IP) we are dealing with is the following: