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Adaptive version of Simpler GMRES

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Abstract

In this paper we propose a stable variant of Simpler GMRES. It is based on the adaptive choice of the Krylov subspace basis at a given iteration step using the intermediate residual norm decrease criterion. The new direction vector is chosen as in the original implementation of Simpler GMRES or it is equal to the normalized residual vector as in the GCR method. We show that such an adaptive strategy leads to a well-conditioned basis of the Krylov subspace and we support our theoretical results with illustrative numerical examples.

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Author notes

  1. Pavel Jiránek

    Present address: CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse Cedex 1, France

Authors and Affiliations

  1. Faculty of Mechatronics, Technical University of Liberec, Studentská 2, 46117, Liberec, Czech Republic

    Pavel Jiránek

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 18207, Prague 8, Czech Republic

    Miroslav Rozložník

Authors
  1. Pavel Jiránek
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  2. Miroslav Rozložník
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Corresponding author

Correspondence to Pavel Jiránek.

Additional information

The work of the first author was supported by the grant No. 201/09/P464 of the Grant Agency of the Czech Republic.

The work of the second author was supported by the project IAA100300802 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Institutional Research Plan AV0Z10300504 “Computer Science for the Information Society: Models, Algorithms, Applications”.

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Jiránek, P., Rozložník, M. Adaptive version of Simpler GMRES. Numer Algor 53, 93–112 (2010). https://doi.org/10.1007/s11075-009-9311-2

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  • DOI: https://doi.org/10.1007/s11075-009-9311-2

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