  We consider a regularization concept for the solution of ill--posed operator
equations, where the operator is composed of a continuous and a discontinuous
operator. A particular application is level set regularization, where we
develop a novel concept of minimizers. The proposed level set regularization is
capable of handling changing topologies. A functional analytic framework
explaining the splitting of topologies is given. The asymptotic limit of the
level set regularization method is an evolution process, which is implemented
numerically and the quality of the proposed algorithm is demonstrated by
solving an inverse source problem.
