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A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations
Author(s) -
Bağcı Hakan,
Pasciak Joseph E.,
Sirenko Kostyantyn Y.
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1961
Subject(s) - preconditioner , mathematics , tridiagonal matrix , gaussian elimination , rank (graph theory) , block matrix , block (permutation group theory) , convergence (economics) , linear system , tridiagonal matrix algorithm , system of linear equations , matrix (chemical analysis) , inverse , iterative method , gaussian , algorithm , mathematical analysis , combinatorics , eigenvalues and eigenvectors , geometry , physics , quantum mechanics , economics , composite material , economic growth , materials science
Summary We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off‐diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.