Solution smoothness of ill-posed equations in Hilbert spaces: four concepts and their cross connections

Numerical solution of ill-posed operator equations requires regularization techniques. The convergence of regularized solutions to the exact solution can be usually guaranteed, but to also obtain estimates for the speed of convergence one has to exploit some kind of smoothness of the exact solution. We consider four such smoothness concepts in a Hilbert space setting: source conditions, approximate source conditions, variational inequalities, and approximate variational inequalities. Besides some new auxiliary results on variational inequalities the equivalence of the last three concepts is shown. In addition, it turns out that the classical concept of source conditions and the modern concept of variational inequalities are connected via Fenchel duality.