A note on adaptivity in factorized approximate inverse preconditioning

Author:

Kopal Jiří1,Rozložník Miroslav2,Tůma Miroslav3

Affiliation:

1. Technical University of Liberec , Institute of Novel Technologies and Applied Informatics , Studentská 1402/2, 461 17 Liberec 1, Czech Republic

2. Institute of Computer Science, Academy of Sciences of the Czech Republic , Pod Vodárenskou věží 2, 182 07 Prague 8 , Czech Republic

3. Faculty of Mathematics and Physics , Charles University in Prague , Sokolovská 83, 186 75 Praha 8 ,

Abstract

Abstract 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 di cult 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.

Publisher

Walter de Gruyter GmbH

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Sparse Approximate Inverse Preconditioners;Nečas Center Series;2023

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