Levenberg–Marquardt level set methods for inverse obstacle problems

Abstract

The aim of this paper is to construct Levenberg–Marquardt level set methods for inverse obstacle problems, and to discuss their numerical realization. Based on a recently developed framework for the construction of level set methods, we can define Levenberg–Marquardt level set methods in a general way by varying the function space used for the normal velocity. In the typical case of a PDE-constraint, the iterative method yields an indefinite linear system to be solved in each iteration step, which can be reduced to a positive definite problem for the normal velocity. We discuss the structure of this system and the possibilities for its iterative solution. Moreover, we investigate the application and numerical discretization of the method for two model problems, a mildly ill-posed source reconstruction problem and a severely ill-posed identification problem from boundary data. The numerical results demonstrate a significant speed-up with respect to standard gradient-based level set methods, in particular if topology changes occur during the level set evolution.