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A Note on Adaptivity in Factorized Approximate Inverse Preconditioning
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SYSNO ASEP 0525243 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title A Note on Adaptivity in Factorized Approximate Inverse Preconditioning Author(s) Kopal, J. (CZ)
Rozložník, Miroslav (UIVT-O) SAI, RID, ORCID
Tůma, M. (CZ)Number of authors 3 Source Title Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica. - : Ovidius University Press - ISSN 1224-1784
Roč. 28, č. 2 (2020), s. 149-159Number of pages 11 s. Language eng - English Country RO - Romania Keywords sparse approximate inverse preconditioners ; approximate factorization ; generalized Gram-Schmidt process Subject RIV BA - General Mathematics OECD category Applied mathematics R&D Projects GA17-12925S GA ČR - Czech Science Foundation (CSF) Method of publishing Open access Institutional support UIVT-O - RVO:67985807 UT WOS 000574556300009 EID SCOPUS 85093503628 DOI 10.2478/auom-2020-0024 Annotation The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram-Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve difficult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram-Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2021 Electronic address https://www.anstuocmath.ro/volume-xxviii-2020-fascicola-2.html
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