Non-Convex Regularization

Inverse Problems have gained significant importance in the last decades in industrial applications, such as non- destructive evaluation and medical imaging. The associated mathematics necessary to solve and analyze inverse problems is a challenging topic. One of the mayor issues in numerically solving inverse problems is to deal with their instable behaviour caused by data and numerical errors. This unstable behaviour is refered to as ill-posedness. This project is concerned with the analysis and implementation of new variational methods for solving inverse problems in a stable way. Such methods are called regularization techniques. The history of regularization methods starts with the piooneering works of A. Tikhonov in the thirties. Later on the theory of linear inverse problems developed systematically until in the eighties a fairly complete analysis became available. Still nowadays efficient (fast) regularization techniques for solving linear inverse problems are a challenging topic. Later on regularization theory for nonlinear inverse and ill-posed problems was developed. In the mid nineties non-differentiable regularization methods came up. A rigorous and complete analysis of regularization methods for non-differentiable regularization and nonlinear inverse problems is far away from being complete and there are many challenging questions. However, the efficiency of such methods is beyond dispute. The project presents a conceptual step beyond non-linearity and non-differentiabilty which is nonconvex regularization. The additional freedom in the concept allows us to construct regularization methods with new properties which could not be achieved so far.