Ronny Ramlau, Lothar Reichel (Hg.)


ETNA - Electronic Transactions on Numerical Analysis






ISBN 978-3-7001-8258-0
Online Edition

  Research Article
Open access


Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Austrian Academy of Sciences Press
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ETNA - Electronic Transactions on Numerical Analysis



ISBN 978-3-7001-8258-0
Online Edition



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
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doi:10.1553/etna_vol55s532



doi:10.1553/etna_vol55s532



Thema: natural
Ronny Ramlau, Lothar Reichel (Hg.)


ETNA - Electronic Transactions on Numerical Analysis






ISBN 978-3-7001-8258-0
Online Edition

  Research Article
Open access


Kirk M. Soodhalter
PDF Icon  A note on augmented unprojected Krylov subspace methods ()
S.  532 - 546
doi:10.1553/etna_vol55s532

Open access

Verlag der Österreichischen Akademie der Wissenschaften


doi:10.1553/etna_vol55s532
Abstract:
Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al., GAMM-Mitt., 43 (2020), Art. e202000016]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R3GMRES method proposed in [Dong et al., Electron., Trans., Numer., Anal., 42 (2014), pp. 136–146] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad, SIAM J. Matrix Anal. Appl., 18 (1997)].

Keywords:  Krylov subspaces, augmentation, recycling, discrete ill-posed problems
  2022/06/02 13:43:13
Object Identifier:  0xc1aa5572 0x003d75a7
.

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.



Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at