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 $K$-interpolation theory. We demonstrate that this leads to simple, self-contained proofs and to rather general results. In particular, we obtain optimal convergence rates for bounded variation regularization, general Besov penalty terms, and $\ell^p$ wavelet penalization with $p<1$, which cannot be treated by previous approaches. We also derive minimax optimal rates for white noise models. Our theoretical results are confirmed in numerical experiments.

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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

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