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A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems

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Abstract

We consider the efficient solution of linear systems with multiple shifts and multiple right-hand sides given simultaneously that arise frequently in large-scale scientific and engineering simulations. We introduce a new shifted block GMRES method that can solve the whole sequence of linear systems simultaneously, it handles effectively the situation of inexact breakdowns in the inner block Arnoldi procedure for improved robustness, and recycles spectral information at restart to achieve faster convergence. Numerical experiments are reported on a suite of sparse matrix problems and in realistic quantum chromodynamics application to show the potential of the new proposed method to solve general multi-shifted and multiple right-hand sides linear systems fast and efficiently.

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  1. https://zenodo.org/record/56157#.WtTbCvknbCA.

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Acknowledgements

This research is supported by NSFC (61772003). The first author is also funded by the University of Groningen Ubbo Emmius scholarship. Finally, we would like to thank the anonymous referees for their valuable remarks, questions, and comments that enabled us to improve this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Dong-Lin Sun, Ting-Zhu Huang & Yan-Fei Jing

  2. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, PO Box 407, 9747 AG, Groningen, The Netherlands

    Dong-Lin Sun

  3. Faculty of Computer Science, Free University of Bozen-Bolzano, Piazza Domenicani, 3, 39100, Bolzano, Italy

    Bruno Carpentieri

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  1. Dong-Lin Sun
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  2. Ting-Zhu Huang
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  3. Bruno Carpentieri
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  4. Yan-Fei Jing
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Correspondence to Ting-Zhu Huang or Yan-Fei Jing.

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The original version of this article was revised with correct figure 1.

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Sun, DL., Huang, TZ., Carpentieri, B. et al. A New Shifted Block GMRES Method with Inexact Breakdowns for Solving Multi-Shifted and Multiple Right-Hand Sides Linear Systems. J Sci Comput 78, 746–769 (2019). https://doi.org/10.1007/s10915-018-0787-6

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