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Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ1-regularization

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Published/Copyright: June 7, 2017

Abstract

We show that the convergence rate of 1-regularization for linear ill-posed equations is always 𝒪(δ) if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.

MSC 2010: 65J20; 47A52

Award Identifier / Grant number: FL 832/1-2

Award Identifier / Grant number: HO 1454/8-2

Award Identifier / Grant number: HO 1454/10-1

Funding statement: Research was supported by DFG grants FL 832/1-2, HO 1454/8-2 and HO 1454/10-1.

Acknowledgements

We thank Bernd Hofmann (TU Chemnitz) for many valuable comments on a draft of this paper and for fruitful discussions on the subject. We also thank the two anonymous referees whoes comments helped to improve readability of the paper.

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Received: 2017-1-13
Revised: 2017-5-5
Accepted: 2017-5-5
Published Online: 2017-6-7
Published in Print: 2018-2-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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