A framework for the construction of level set methods for shape optimization and reconstruction

The aim of this paper is to develop a functional-analytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient flows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case. Moreover, we apply this framework to some (inverse) model problems for elliptic boundary value problems. In numerical experiments we demonstrate that an appropriate choice of norms (dependent on the problem) yields stable and fast methods.