  We propose a numerical algorithm for the reconstruction of a piecewise
constant leading coefficient of an elliptic problem. The inverse problem is
reduced to a shape reconstruction problem. The proposed algorithm is based on
the minimization of a cost functional where a control function is the
right-hand side of an auxiliary elliptic equation for a level set
representation of unknown shape. The numerical implementation is based on the
finite element method and the open-source computing platform FEniCS. The
performance of the algorithm is demonstrated on computationally simulated data.
