An overview on convergence rates for Tikhonov regularization methods for non-linear operators

Abstract

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. The first convergence rates results for non-linear problems have been developed by Engl, Kunisch and Neubauer in 1989 [3]. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004 [1], more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and Scherzer [6] presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions on the operator do not necessarily affect the convergence rate negatively. We have taken this observation and weakened the smoothness assumptions on the operator and have proved a novel convergence rate result published in [4]. The most significant difference of this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before, which is necessary due to the lack of smoothness assumptions on F .

Generalized Bregman distances and convergence rates for non-convex regularization methods

“…In addition, corollary 3.1 yields convergence rates for a parameter choice that depends on the behavior of the similarity term near F (x † ). For the case of metric regularization, where the similarity term is some power of the distance on the target space Y, these rates reduce precisely to the ones derived in [14,16,20] (see example 3.1).…”

“…In case we have no uniqueness, we denote by x δ α any minimizer of T α (•, y δ ). Collecting the results of [14,16,20], we see that the minimization of T α is a welldefined regularization method (that is, it attains a solution that is stable with respect to data perturbations and converges to the true solution as the noise level decreases to zero), if the following conditions are satisfied for some topologies on X and Y.…”

“…The stronger assumption (A1), however, will be required below for the derivation of convergence rates. More details on the Tikhonov regularization in such a general setting can be found in [16].…”

“…This generalization uses an abstraction of the notion of convexity that relies on more general dualities than the one between a Banach space X and the Banach space X * of all bounded linear functionals on X. Moreover, generalizing the results of [16], we consider more general similarity terms than the squared norm of the residual.…”

Minisymposium — Recent progress in regularization theory

“…Christiane Pöschl, A convergence rates result in Banach spaces with non-smooth operators (see [11]). …”