Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Abstract

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.

Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

“…By Assumption 6.1 we have N := g obs | B −d/2 p ′ ,∞ < ∞ almost surely. As in the proof of [39,Prop. 4.8] we can bound the data fidelity term from below using eqs.…”

“…We start from lower bounds for the estimation of g † = F (f † ), see [12,Thms. 7,9], [39,Cor. 4.12].…”

“…(Recall that source conditions in regularization theory are conditions which allow to control the bias term/approximation error, see Section 3 for details.) This adapts a strategy based on a sequence of finite-dimensional subspaces and associated semi-norms which was proposed in [24,39] for the case of smooth penalties. We show that our strategy is also a generalization of that in [2].…”

“…• Finally, in Theorem 6.4 we derive convergence rates for the white noise model (4) using a method proposed in [39]. These rates are optimal for p > 1 and almost optimal for p = 1.…”

Convergence analysis of (statistical) inverse problems under conditional stability estimates

“…More recently, there have been approaches (see e.g. [23,24,25,35]) to verify variational source conditions directly for specific problem instances without relying on nonlinearity assumptions or spectral source conditions or on both. In this case, it can happen that the set M allows to interchange f and f † in (11), and hence also a conditional stability estimate follows, see e.g.…”