A level set method for inverse problems

This paper is devoted to the solution of shape reconstruction problems by a level set method. The basic motivation for the setup of this level set algorithm is the well-studied method of asymptotic regularization, which has been developed for ill-posed problems in Hilbert spaces. Using analogies to this method, the convergence analysis of the proposed level set method is established and it is shown that the evolving level set converges to a solution in the symmetric difference metric as the artificial time evolves to infinity. Furthermore, the regularizing properties of the level set method are shown, if the discrepancy principle is used as a stopping rule. The numerical implementation of the level set method is discussed and applied to some examples in order to compare the numerical results with theoretical statements. The numerical results demonstrate the power of the level set method, in particular for examples where the number of connected components the solution consists of is not known a priori.