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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year


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ISSN
1569-3945
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Volume 18, Issue 6 (Jan 2010)

Issues

Discrepancy curves for multi-parameter regularization

Shuai Lu
  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergstrasse 69, 4040 Linz, Austria and School of Mathematical Science, Fudan University, 200433 Shanghai, P.R. China.
  • Email:
/ Sergei V. Pereverzev
  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergstrasse 69, 4040 Linz, Austria.
  • Email:
/ Yuanyuan Shao
  • Department of Mathematics, University of Applied Sciences Zittau/Görlitz, P.O. Box 1455, 02755 Zittau, Germany.
  • Email:
/ Ulrich Tautenhahn
  • Department of Mathematics, University of Applied Sciences Zittau/Görlitz, P.O. Box 1455, 02755 Zittau, Germany.
  • Email:
Published Online: 2010-12-20 | DOI: https://doi.org/10.1515/jiip.2010.030

Abstract

For solving linear ill-posed problems regularization methods are required when the right-hand side is with some noise. In the present paper regularized solutions are obtained by multi-parameter regularization and the regularization parameters are chosen by a multi-parameter discrepancy principle. Under certain smoothness assumptions we provide order optimal error bounds that characterize the accuracy of the regularized solutions. For the computation of the regularization parameters fast algorithms of Newton type are applied which are based on special transformations. These algorithms are globally and monotonically convergent. Some of our theoretical results are illustrated by numerical experiments. We also show how the proposed approach may be employed for multi-task approximation.

Keywords.: Ill-posed problems; inverse problems; noisy right-hand side; Tikhonov regularization; multi-parameter regularization; discrepancy principle; order optimal error bounds; Newton's method; global convergence; monotone convergence

About the article

Received: 2010-03-30

Published Online: 2010-12-20

Published in Print: 2010-12-01



Citation Information: Journal of Inverse and Ill-posed Problems, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip.2010.030. Export Citation

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[1]
Herbert Egger, Jan-Frederik Pietschmann, and Matthias Schlottbom
Inverse Problems, 2014, Volume 30, Number 2, Page 025004

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