Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient (Q2806032)

The authors consider the inverse problem of the recovery of the piecewise constant diffusion coefficients for a Neumann boundary value problem attached to a linear elliptic equation. The functional used in optimal control contains a usual quadratic fidelity term as well as a newly added one, i.e., a perimeter regularization weighted by a specified parameter. For computational purposes the perimeter functional is upgraded using a gradient energy functional together with an obstacle potential. The finite element solution of the phase field equation is proved to be convergent. An iterative algorithm providing an energy decreasing and converging sequence to a critical point is introduced and analyzed. Some 2D numerical experiments are carried out in order to illustrate the efficiency of the proposed algorithm.