Startseite Mathematik Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints
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Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

  • Wei Wang , Shuai Lu EMAIL logo , Bernd Hofmann ORCID logo und Jin Cheng
Veröffentlicht/Copyright: 13. Juni 2019

Abstract

Measuring the error by an 1-norm, we analyze under sparsity assumptions an 0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations Ax=y with an injective and bounded linear operator A mapping between 2 and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the 0-term and the complementing convex penalty, the important special case of the 2-norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

MSC 2010: 65J20; 47A52

Award Identifier / Grant number: 11401257

Award Identifier / Grant number: 91730304

Award Identifier / Grant number: 11331004

Award Identifier / Grant number: 11421110002

Award Identifier / Grant number: LY19A010009

Award Identifier / Grant number: 16SG01

Award Identifier / Grant number: HO 1454/12-1

Funding statement: W. Wang is supported by NSFC (No. 11401257) and Natural Science Foundation of Zhejiang Province (No. LY19A010009). S. Lu is supported by NSFC (No. 91730304), Shanghai Municipal Education Commission (No. 16SG01), Program of Shanghai Academic/Technology Research Leader (19XD1420500) and National Key Research and Development Program of China (No. 2017YFC1404103). B. Hofmann is supported by Deutsche Forschungsgemeinschaft under DFG-grant HO 1454/12-1. J. Cheng is supported by NSFC (No. 11331004, No. 11421110002) and the Programme of Introducing Talents of Discipline to Universities (No. B08018).

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Received: 2019-01-23
Revised: 2019-04-19
Accepted: 2019-04-23
Published Online: 2019-06-13
Published in Print: 2019-08-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2019-0008/pdf
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