  We consider the inverse problem of recovering a binary function from blurred
and noisy data. Such problems arise in many applications, for example image
processing and optimal control of PDEs. Our formulation is based on the
Mumford-Shah model, but with a phase field approximation to the perimeter
regularisation. We use a double obstacle potential as well as a smooth double
well potential. We introduce an iterative method for solving the problem,
develop a suitable discretisation of this iterative method, and prove some
convergence results. Numerical simulations are presented which illustrate the
usefulness of the approach and the relative merits of the phase field models.

Optimization and Control