  We study the convergence of variationally regularized solutions to linear
ill-posed operator equations in Banach spaces as the noise in the right hand
side tends to $0$. The rate of this convergence is determined by abstract
smoothness conditions on the solution called source conditions. For
non-quadratic data fidelity or penalty terms such source conditions are often
formulated in the form of variational inequalities. Such variational source
conditions (VSCs) as well as other formulations of such conditions in Banach
spaces have the disadvantage of yielding only low-order convergence rates. A
first step towards higher order VSCs has been taken by Grasmair (2013) who
obtained convergence rates up to the saturation of Tikhonov regularization. For
even higher order convergence rates, iterated versions of variational
regularization have to be considered. In this paper we introduce VSCs of
arbitrarily high order which lead to optimal convergence rates in Hilbert
spaces. For Bregman iterated variational regularization in Banach spaces with
general data fidelity and penalty terms, we derive convergence rates under
third order VSC. These results are further discussed for entropy regularization
with elliptic pseudodifferential operators where the VSCs are interpreted in
terms of Besov spaces and the optimality of the rates can be demonstrated. Our
theoretical results are confirmed in numerical experiments.
