A Note on Adaptivity in Factorized Approximate Inverse Preconditioning

0525243 - ÚI 2021 RIV RO eng J - Journal Article
Kopal, J. - Rozložník, Miroslav - Tůma, M.
A Note on Adaptivity in Factorized Approximate Inverse Preconditioning.
Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica. Roč. 28, č. 2 (2020), s. 149-159. ISSN 1224-1784. E-ISSN 1844-0835
R&D Projects: GA ČR(CZ) GA17-12925S
Institutional support: RVO:67985807
Keywords : sparse approximate inverse preconditioners * approximate factorization * generalized Gram-Schmidt process
OECD category: Applied mathematics
Impact factor: 1.045, year: 2020
Method of publishing: Open access
https://www.anstuocmath.ro/volume-xxviii-2020-fascicola-2.html

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.
Permanent Link: http://hdl.handle.net/11104/0309424