Incorporating topological derivatives into level set methods

The aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems. We propose a new approach generalizing the standard speed method, which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional. The resulting approach can be interpreted as a generalized fixed-point iteration for the optimality system (with respect to topological and shape variations). Moreover, we apply the new approach for a simple model problem in shape reconstruction, where the topological derivative can be computed without additional effort. Finally, we present numerical tests related to this model problem, which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails.