Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces (Q2848401)

There is a well-established theory of convergence rates in variational (Tikhonov-type) regularization for ill-posed linear operator equations in Hilbert spaces with quadratic misfit and penalty terms. However, if the theory is to be transferred to Banach spaces or/and to general convex penalties, essential new difficulties occur and new mathematical tools are required. The author provides in the introduction of this paper an excellent overview with relevant references of the new challenges of this transition and of the progress in this field over the last ten years. A very successful new technique is based on the use of variational inequalities for the description of the solution smoothness with respect to the occurring forward operator (cf., e.g., Section 3.2 of the monograph by \textit{O. Scherzer} et al. [Variational methods in imaging. New York, NY: Springer (2009; Zbl 1177.68245)] and Section 3.2.3 of the monograph by \textit{T. Schuster} [Regularization methods in Banach spaces. Berlin: de Gruyter (2012; Zbl 1259.65087)]). This technique supplements and partly replaces the usually considered source conditions for expressing the solution smoothness. The present paper is the first step in order to achieve rates of higher order based on variational inequalities measured by the Bregman distance. All previous papers on convergence rates for linear ill-posed problems exploiting variational inequalities were limited to low rate results. The author successfully applies minimizers of the dual Tikhonov functional in combination with variational inequalities for the Bregman distance of the conjugate functional of the convex penalty. In this way, he reaches the same higher order convergence rates which were proved in the article by \textit{A. Neubauer} et al. [Appl. Anal. 89, No. 11, 1729--1743 (2010; Zbl 1214.65031)] by using a completely different approach. He thus refutes the conjecture that variational inequalities are only helpful for obtaining low order rates. It seems to be an open question whether the technique developed by the author can be extended to higher order rates for nonlinear ill-posed problems.