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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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Discrepancy curves for multi-parameter regularization

1Johann 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.

2Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergstrasse 69, 4040 Linz, Austria.

3Department of Mathematics, University of Applied Sciences Zittau/Görlitz, P.O. Box 1455, 02755 Zittau, Germany.

4Department of Mathematics, University of Applied Sciences Zittau/Görlitz, P.O. Box 1455, 02755 Zittau, Germany.

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 18, Issue 6, Pages 655–676, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/jiip.2010.030, December 2010

Publication History

Received:
2010-03-30
Published Online:
2010-12-20

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

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