  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 quadratic fidelity term to which we add a perimeter
regularisation weighted by a parameter sigma. This yields a functional which is
optimised over a set of diffusion coefficients subject to a state equation
which is the underlying elliptic PDE. In order to derive a problem which is
amenable to computation the perimeter functional is relaxed using a gradient
energy functional together with an obstacle potential in which there is an
interface parameter epsilon. This phase field approach is justified by proving
Gamma-convergence to the functional with perimeter regularisation as epsilon
tends to zero. The computational approach is based on a finite element
approximation. This discretisation is shown to converge in an appropriate way
to the solution of the phase field problem. We derive an iterative method which
is shown to yield an energy decreasing sequence converging to a discrete
critical point. The efficacy of the approach is illustrated with numerical
experiments.
