Shape sensitivity analysis for an interface problem via minimax differentiability

Abstract

This paper deals with the shape sensitivity analysis for the conductivity problem with discontinuous coefficients. We consider the geometric inverse problem of locating the interfaces where the jump of the coefficients occurs. We propose a reconstruction method based on the minimization of the Kohn–Vogelius functional. The reformulation of the functional in a suitable saddle point problem form allows us to obtain the optimality conditions by using the differentiability of the minimax combined with the function space parametrization technique. An important advantage of this approach is that it avoids the differentiability of the state variable with respect to the domain. The reconstruction is then performed with an iterative algorithm using an appropriate Quasi-Newton method. We give some numerical examples showing the efficiency of the method.