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This paper addresses Tikhonov like regularization methods with convex penaltyfunctionals for solving nonlinear ill-posed operator equations formulated inBanach or, more general, topological spaces. We present an approach for provingconvergence rates which combines advantages of approximate source conditionsand variational inequalities. Precisely, our technique provides both a widerange of convergence rates and the capability to handle general and notnecessarily convex residual terms as well as nonsmooth operators. Initiallyformulated for topological spaces, the approach is extensively discussed forBanach and Hilbert space situations, showing that it generalizes somewell-known convergence rates results.

A new approach to source conditions in regularization with general residual term