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On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems

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Abstract

The problem of shifted linear systems is an important and challenging issue in a number of research applications. Krylov subspace methods are effective techniques for different kinds of this problem due to their advantages in large and sparse matrix problems. In this paper, two new block projection methods based on respectively block FOM and block GMRES are introduced for solving sequences of shifted linear systems. We first express the original problem explicitly by a sequence of Sylvester matrix equations whose coefficient matrices are obtained from the shifted linear systems. Then, we show the restarted shifted block FOM (rsh-BFOM) method and derive some of its properties. We also present a framework for the restarted shifted block GMRES (rsh-BGMRES) method. In this regard, we describe two variants of rsh-BGMRES, including (1) rsh-BGMRES with an unshifted base system that applies a fixed unshifted base system and (2) rsh-BGMRES with a variable shifted base system in which the base block system can change after restart. Furthermore, we consider the use of deflation techniques for improving the performance of the rsh-BFOM and rsh-BGMRES methods. Finally, some numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

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Notes

  1. When block vectors are used, some authors prefer to use the word cospatial instead of collinear [42, 43]

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Authors and Affiliations

  1. Centre Régional des Métiers de l’Education et de la Formation, Fes, Morocco

    Lakhdar Elbouyahyaoui

  2. L.M.P.A, Université du Littoral Côte d’Opale, 50 rue F. Buisson BP. 699, F-62228 Calais Cedex, Dunkirk, France

    Mohammed Heyouni

  3. Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Azita Tajaddini

  4. Department of Applied Mathematics, Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran

    Farid Saberi-Movahed

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  1. Lakhdar Elbouyahyaoui
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Elbouyahyaoui, L., Heyouni, M., Tajaddini, A. et al. On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems. Numer Algor 87, 1257–1299 (2021). https://doi.org/10.1007/s11075-020-01007-3

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