Methods for reliable topology changes for perimeter-regularized geometric inverse problems

This paper is devoted to the incorporation of topological derivativelike expansions into level set methods for perimeter-regularized geometric inverse problems. The expansions are done up to the second order with respect to the Lebesgue measure of the symmetric difference. They provide simpler shape functionals, still including the perimeter, and therefore allow the construction of steepest descent- and Newton-type algorithms to force topology changes during the level set evolution. Numerous numerical examples are provided that show the strong and also the weak points of the newly developed algorithms.