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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 25, Issue 5


A conjugate direction method for linear systems in Banach spaces

Roland HerzogORCID iD: http://orcid.org/0000-0003-2164-6575 / Winnifried WollnerORCID iD: http://orcid.org/0000-0002-6571-8043
Published Online: 2016-11-17 | DOI: https://doi.org/10.1515/jiip-2016-0027


In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

Keywords: Conjugate direction method; reflexive Banach space; linear operator equation

MSC 2010: 65J10; 65J22; 65F25


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About the article

Received: 2016-04-22

Revised: 2016-10-13

Accepted: 2016-10-31

Published Online: 2016-11-17

Published in Print: 2017-10-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 5, Pages 553–572, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2016-0027.

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