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ETNA - Electronic Transactions on Numerical Analysis
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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 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 206-230, 2018/02/13
Krylov subspace methods are commonly used iterative methods for solving largesparse linear systems. However, they suffer from communication bottlenecks onparallel computers.Therefore, $s$-step methods have been developed, where the Krylov subspace isbuilt block by block so that $s$ matrix-vector multiplications can be donebefore orthonormalizing the block.Then Communication-Avoiding algorithms can be used for both kernels.This paper introduces a new variation on the $s$-step GMRES method in order toreduce the number of iterations necessary to ensure convergence with a smalloverhead in the number of communications.Namely, we develop an $s$-step GMRES algorithm, where the block size isvariable and increases gradually.Our numericalexperiments show a good agreement with our analysis of condition numbersand demonstratethe efficiency of our variable $s$-step approach.
Keywords: Communication-Avoiding, s-step Krylov subspace method, GMRES algorithm, variable s-step