Classical results for quadratic regularisation in Hilbert spaces show convergence rates with respect to the norm under the assumption that the true solution satisfies a source condition in the sense that it is an element of some fractional power of the operator to be inverted. These results have later been generalised to non-quadratic regularisation terms, though with two significant changes: First, the norm had to be replaced by the Bregman distance for the regularisation term, particularly to be able to deal with terms that fail to be strictly convex. Second, although the first results were derived using basic source conditions, these were later replaced by variational inequalities in order to obtain sharper and more general estimates, and also to include Banach space settings.
In this talk we return to the study of source conditions for non-quadratic problems on Hilbert spaces. We will derive convergence rates with respect to the Bregman distance under the assumptions of uniform convexity of either the regularisation term or its convex conjugate and satisfaction of a Hölder type source condition. In the quadratic case, both the conditions and the convergence rates turn out to be identical to the classical results.
