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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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

Wei Wang
/ Shuai Lu
• Corresponding author
• Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China
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• Other articles by this author:
/ Bernd Hofmann
/ Jin Cheng
• Shanghai Key Laboratory for Contemporary Applied Mathematics and School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China
• Email
• Other articles by this author:
Published Online: 2019-06-13 | DOI: https://doi.org/10.1515/jiip-2019-0008

## Abstract

Measuring the error by an ${\mathrm{\ell }}^{1}$-norm, we analyze under sparsity assumptions an ${\mathrm{\ell }}^{0}$-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 ${\mathrm{\ell }}^{2}$ 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 ${\mathrm{\ell }}^{0}$-term and the complementing convex penalty, the important special case of the ${\mathrm{\ell }}^{2}$-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

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Revised: 2019-04-19

Accepted: 2019-04-23

Published Online: 2019-06-13

Published in Print: 2019-08-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11401257

Award identifier / Grant number: 91730304

Award identifier / Grant number: 11331004

Award identifier / Grant number: 11421110002

Funding Source: Natural Science Foundation of Zhejiang Province

Award identifier / Grant number: LY19A010009

Funding Source: Shanghai Municipal Education Commission

Award identifier / Grant number: 16SG01

Funding Source: Deutsche Forschungsgemeinschaft

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

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).

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 4, Pages 575–590, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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