Philip Miller,
Thorsten Hohage
S. 101 - 126
doi:
10.1553/etna_vol57s101
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol57s101
Abstract:
This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalizationenforces too much smoothness in the sense that the penalty term is not finite at the truesolution. 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 progressbeen 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 familyof smooth approximations based on K-interpolation theory. We demonstrate that thisleads to simple, self-contained proofs and to rather general results.In particular, we obtain optimal convergencerates for bounded variation regularization, general Besov penalty terms, and $\ell^p$ wavelet penalization with $p<1$, which cannotbe treated by previous approaches. We also derive minimax optimal rates for white noisemodels. Our theoretical results are confirmed in numerical experiments.
regularization, convergence rates, oversmoothing, BV-regularization, sparsity-promoting wavelet regularization, statistical inverse problems
Published Online:
2022/06/27 09:50:37
Object Identifier:
0xc1aa5576 0x003d84a1
Rights:All rights reserved.For questions regarding copyright and copies please contact us by email.
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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