Volume 57, pp. 101-126, 2022.
Convergence rates for oversmoothing Banach space regularization
Philip Miller and Thorsten Hohage
Abstract
This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalization
enforces too much smoothness in the sense that the penalty term is not finite at the true
solution. In a Hilbert space setting, Natterer [Applicable Anal., 18 (1984),
pp. 29–37] showed with the help of spectral theory that optimal rates can be achieved in this situation. (“Oversmoothing does not harm.”)
For oversmoothing variational regularization in Banach spaces, only very recently has progress
been achieved in several papers in different settings, all of which construct families of smooth approximations to the true solution. In this paper we propose to construct such a family
of smooth approximations based on
Full Text (PDF) [439 KB], BibTeX , DOI: 10.1553/etna_vol57s101
Key words
regularization, convergence rates, oversmoothing, BV-regularization, sparsity-promoting wavelet regularization, statistical inverse problems
AMS subject classifications
65J22, 65N21, 35R20
Links to the cited ETNA articles
[16] | Vol. 53 (2020), pp. 313-328 Bernd Hofmann and Robert Plato: Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term |