Levenberg–Marquardt level set methods for inverse obstacle problems

Abstract

The aim of this paper is to construct Levenberg–Marquardt level set methods for inverseobstacle problems, and to discuss their numerical realization. Based on a recentlydeveloped framework for the construction of level set methods, we can defineLevenberg–Marquardt level set methods in a general way by varying the function spaceused for the normal velocity. In the typical case of a PDE-constraint, the iterativemethod 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. Wediscuss the structure of this system and the possibilities for its iterative solution.Moreover, we investigate the application and numerical discretization of the methodfor two model problems, a mildly ill-posed source reconstruction problem and aseverely ill-posed identification problem from boundary data. The numerical resultsdemonstrate a significant speed-up with respect to standard gradient-based level setmethods, in particular if topology changes occur during the level set evolution.