Shape optimization methods for detecting an unknown boundary with the Robin condition by a single boundary measurement

We consider the problem of identifying an unknown portion $ \Gamma $ of the boundary of a $ d $-dimensional ($ d = 2, 3 $) body $ \Omega $ by a pair of Cauchy data $ (f, g) $ on the accessible part $ \Sigma $ of the boundary of a harmonic function $ u $. On the unknown boundary, a Robin homogeneous condition is assumed. For a fixed constant impedance $ \alpha $, it was shown by Cakoni and Kress in [1] through concrete examples that a single measurement of $ (f, g) $ on $ \Sigma $ can give rise to infinitely many different domains $ \Omega $. Nonetheless, shape optimization techniques can provide fair detections of the unknown boundary given a single pair of Cauchy data as we will showcase here. On this purpose, the inverse problem is recast into three different shape optimization formulations and the shape derivative of the cost function associated with each formulations are obtained. The shape gradient informations are then utilized in a Sobolev gradient-based scheme via finite element method to solve the optimization problems. Numerical results are provided to illustrate the feasibility of the proposed numerical methods in two and three spatial dimensions.