Regularization of the inverse medium problem : on nonstandard methods for sparse reconstruction

Abstract

In this thesis, we investigate nonstandard methods for the stable solution of the inverse medium problem. Particularly, we consider the linearization of the model of the scattering process given by the Born approximation and investigate regularization methods that are designed for sparse reconstruction. In numerical experiments we demonstrate that sparsity constraints contribute to meaningful reconstructions from synthetic and even measurement data. In our investigations, we consider both iterative and variational methods for the solution of the inverse problem. Starting from the Landweber iteration, we discuss existing variants of this approach and develop a novel sparsity-enforcing method which is based on the Bregman projection. Furthermore, we consider a variational regularization scheme. First, we develop a novel parameter choice rule based on the L-curve criterion designed for sparse reconstruction. We then propose to replace the variational problem by some smooth approximation and provide an exhaustive investigation regarding stability of this approach. The theoretical investigations of each of the methods proposed in this work are complemented by a numerical evaluation.