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Regularization of ill-posed operator equations: an overview

Regularization of ill-posed operator equations: an overview We discuss and illustrate the stability issues associated with ill-posed linear operator equations and the need to have regularization methods for obtaining stable approximate solutions. As illustration, we describe two simple regularization methods, namely, the Lavrentiev regularization in the setting of a Banach spaces and Tikhonov regularization in the setting of Hilbert spaces, and discuss on the corresponding error estimates. We also indicate procedures to obtain error estimates under milder general source conditions and also to obtain better error estimates under modified methods. The discussed procedures include some of the work of the author as well. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

Regularization of ill-posed operator equations: an overview

The Journal of Analysis , Volume 29 (2) – Jul 28, 2020

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Publisher
Springer Journals
Copyright
Copyright © Forum D'Analystes, Chennai 2020
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-020-00263-9
Publisher site
See Article on Publisher Site

Abstract

We discuss and illustrate the stability issues associated with ill-posed linear operator equations and the need to have regularization methods for obtaining stable approximate solutions. As illustration, we describe two simple regularization methods, namely, the Lavrentiev regularization in the setting of a Banach spaces and Tikhonov regularization in the setting of Hilbert spaces, and discuss on the corresponding error estimates. We also indicate procedures to obtain error estimates under milder general source conditions and also to obtain better error estimates under modified methods. The discussed procedures include some of the work of the author as well.

Journal

The Journal of AnalysisSpringer Journals

Published: Jul 28, 2020

Keywords: Ill-posed problems; Least-square solution; Regularization; Hilbert scales; Backward heat equation; Source condition; Lavrentiev; Tikhonov; Morozov; Arcangeli; Optimal rate; 65J10; 65R30; 45B05; 45E99

References