Levenberg–Marquardt level set methods for inverse obstacle problems

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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 deflne 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 indeflnite linear system to be solved in each iteration step, which can be reduced to a positive deflnite problem for the normal velocity. We discuss the structure of this systems and 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 identiflcation problem from boundary data. The numerical results demonstrate a signiflcant speed-up with respect to standard gradient-based level set methods, in par- ticular if topology changes occur during the level set evolution.

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