Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

This paper deals with Tikhonov regularization for linear and nonlinear illposed operator equations with wavelet Besov norm penalties. We focus on B 0 p,1 penalty terms which yield estimators that are sparse with respect to a wavelet frame. Our framework includes among others, the Radon transform and some nonlinear inverse problems in differential equations with distributed measurements. Using variational source conditions it is shown that such estimators achieve minimax-optimal rates of convergence for finitely smoothing operators in certain Besov balls both for deterministic and for statistical noise models.

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

Section: Lower Boundsmentioning

confidence: 99%

“…• 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.…”

Section: Introductionmentioning

confidence: 99%

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

Hohage

Miller

2019

Inverse Problems

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“…Let us mention that variational source conditions, which have been utilized recently for the analysis of regularization methods for nonlinear inverse problems, also allow for the proof of convergence rates without additional assertions on the nonlinearity of the problem. In this context, we refer to [7] and [17] for discussions of cross connections between conditional stability estimates and variational source conditions.…”

Section: Introductionmentioning

confidence: 99%

Tikhonov regularization in Hilbert scales under conditional stability assumptions

Egger

Hofmann

2018

Inverse Problems

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Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a-priori and a-posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.

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“…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.…”

Section: Assumptionmentioning

confidence: 99%

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

Werner

Hofmann

2019

Inverse Problems

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Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems in Hilbert scales satisfying conditional stability estimates characterized by general concave index functions. For that case, we exploit Tikhonov regularization and provide convergence and convergence rates of regularized solutions for both deterministic and stochastic noise. We further discuss a priori and a posteriori parameter choice rules and illustrate the validity of our assumptions in different model and real world situations.

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Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Cited by 4 publications

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

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

Tikhonov regularization in Hilbert scales under conditional stability assumptions

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

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