We consider the identification of a difiusion parameter in a second order elliptic equation in two dimensions by interior measurements. The difiusion parameter is assumed to have discontinuities. For its reconstruction we propose two regularization algorithms–one is based on a moving grid approach and the other on a Hamilton-Jacobi type equation. The numerical realization via an inexact iteratively regularized Gauss-Newton method shows that these algorithms can effectively identify the discontinuities.