Levenberg–Marquardt level set methods for inverse obstacle problems

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 LevenbergMarquardt 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 systems and possibilities for its iterative solution.