Theory and examples of variational regularization with non-metric fitting functionals

We describe and analyze a general framework for solving ill-posed operator equations by minimizing Tikhonov-like functionals. The fitting functional may be non-metric and the operator is allowed to be non-linear and non-smooth. In comparison to former results on variational regularization with non-metric fitting functionals we significantly weaken the assumptions for proving convergence rates and, in addition, we extend the results to a wider range of rates. Two examples, coming from imaging applications, show that the developed theory is applicable to practically relevant problems.