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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 101-126, 2022/06/27
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.
Keywords: regularization, convergence rates, oversmoothing, BV-regularization, sparsity-promoting wavelet regularization, statistical inverse problems