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

Issues

Conjugate gradient regularization under general smoothness and noise assumptions

Gilles Blanchard
  • Universität Potsdam, Institut für Mathematik, Am Neuen Palais 10, 14469 Potsdam, Germany.
  • Email:
/ Peter Mathé
  • Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany.
  • Email:
Published Online: 2010-12-20 | DOI: https://doi.org/10.1515/jiip.2010.033

Abstract

We study noisy linear operator equations in Hilbert space under a self-adjoint operator. Approximate solutions are sought by conjugate gradient type iteration, given as Krylov-subspace minimizers under a general weight function. Solution smoothness is given in terms of general source conditions. The noise may be controlled in stronger norm. We establish conditions under which stopping according to a modified discrepancy principle yields optimal regularization of the iteration. The present analysis extends much of the known theory and reveals some intrinsic features which are hidden when studying standard conjugate gradient type regularization under standard smoothness assumptions. In particular, under a non-self adjoint operator, regularization of the associated normal equation is a direct consequence from the main result and does not require a separate treatment.

Keywords.: Conjugate gradient type regularization; general smoothness

About the article

Received: 2010-07-09

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.2010.033. Export Citation

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[1]
Qinian Jin and Peter Mathé
SIAM/ASA Journal on Uncertainty Quantification, 2013, Volume 1, Number 1, Page 386
[2]
[3]
Bernd Hofmann and Peter Mathé
Inverse Problems, 2012, Volume 28, Number 10, Page 104006
[4]
Peter Mathé and Ulrich Tautenhahn
Journal of Inverse and Ill-posed Problems, 2011, Volume 19, Number 6
[5]
Peter Mathé and Ulrich Tautenhahn
Inverse Problems, 2011, Volume 27, Number 3, Page 035016

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