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
