Recent Developments in Inverse Problems - Abstract

Kindermann, Stefan

New conditions for convergence rates of convex Tikhonov regularization in Banach spaces

We consider Tikhonov regularization in Banach spaces with linear operators and convex regularization terms and convex fidelity terms. [ T(x):= varphi(A x - y^delta) alpha J(x). ] Minimizers of this functionals, $x_alpha,delta,$ are regularized solutions, and we study rates in the Bregman distance for the convergence of $x_alpha,delta$ to a true solution $x^dagger$. For convergence rates, it is necessary that some smoothness condition for $x^dagger$ is satisfied. In a general context, these are nowadays formulated in terms of variational inequalities. For example, if the following condition (“variational inequality”) with some model function $Phi$ holds, [ beta B(x^dagger;x) leq J(x) - J(x^dagger) Phi(A (x^dagger- x)), ] for all $x$ sufficiently close to $x^dagger$ and with some $beta >0$, where $B$ is the Bregman distance, convergence rates have been proved.
The aim of this talk is to report results on convergence rates under a weaker condition, namely assuming that a variational inequality with $beta =0$ holds. In this case we can prove the same convergence rates as those established in the literature.