by Klaus Deckelnick, Charles M. Elliott, Vanessa Styles.

**Series:** 2015-09, Preprints

- MSC:
- 35R30 Inverse problems
- 49N45 Inverse problems

**Abstract:**

We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a fidelity term to be optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. First, a perimeter regularisation weighted by a regularisation parameter is added to a quadratic fidelity functional. Then the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter. The approach is justified by using Gamma-convergence and by finite element calculations. We derive an iterative method which is shown to converge for the discrete optimisation problem.

**Keywords:**

geometric inverse problem, phase field method, finite element approximation