A note on adaptivity in factorized approximate inverse preconditioning

Jiří Kopal 1 , Miroslav Rozložník 2 , and Miroslav Tůma 3
  • 1 Technical University of Liberec, Institute of Novel Technologies and Applied Informatics, Studentská 1402/2, 461 17, Liberec
  • 2 Institute of Computer Science, Academy of Sciences of the Czech Republic, , Pod Vodárenskou věží 2, 182 07, Prague 8
  • 3 Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8

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.

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  • [1] M. Benzi. A Direct Row-Projection Method for Sparse Linear Systems. PhD. Thesis. Department of Mathematics. North Carolina State University, Raleigh, NC, 1993.

  • [2] M. Benzi, J. K. Cullum, and M. Tůma. Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput., 22(4):1318–1332, 2000.

  • [3] M. Benzi, R. Kouhia, and M. Tůma. An assessment of some preconditioning techniques in shell problems. Communications in Numerical Methods in Engineering, 14:897–906, 1998.

  • [4] M. Benzi, R. Kouhia, and M. Tůma. Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics. Comput. Methods Appl. Mech. Engrg., 190(49-50):6533–6554, 2001.

  • [5] M. Benzi, C. D. Meyer, and M. Tůma. A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput., 17(5):1135–1149, 1996.

  • [6] M. Challacombe. A simplified density matrix minimization for linear scaling self-consistent field theory. J. Chem. Phys., 110:2332–2342, 1999.

  • [7] M. Challacombe. A general parallel sparse-blocked matrix multiply for linear scaling scf theory. Comp. Phys. Comm., 128:93–107, 2000.

  • [8] T. Davis and Y. Hu. The university of florida sparse matrix collection. ACM Transactions on Mathematical Software, 38 (1):1–25, 2011.

  • [9] J. Duintjer Tebbens and M. Tůma. On incremental condition estimators in the 2-norm. SIAM J. Matrix Anal. Appl., 35(1):174–197, 2014.

  • [10] N. J. Higham. Accuracy and stability of numerical algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2002.

  • [11] J. Kopal. Generalized Gram–Schmidt Process: Its Analysis and Use in Preconditioning. PhD thesis, Technical University in Liberec, 2014.

  • [12] J. Kopal, M. Rozložník, and M. Tůma. Factorized approximate inverses with adaptive dropping. SIAM J. Sci. Comput., 38(3):A1807–A1820, 2016.

  • [13] M. Rozložník, M. Tůma, A. Smoktunowicz, and J. Kopal. Rounding error analysis of orthogonalization with a non-standard inner product. BIT Numer Math, 52:1035–1058, 2012.

  • [14] E. H. Rubensson, A. G. Artemov, A. Kruchinina, and E. Rudberg. Localized inverse factorization. arXiv:1812.04919.

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