An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices

Xi, Yuanzhe; Li, Ruipeng; Saad, Yousef

doi:10.1137/15m1021830

Title: An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices

Journal Article · Thu Mar 03 00:00:00 EST 2016 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/15m1021830· OSTI ID:1756735

Xi, Yuanzhe ^[1]; Li, Ruipeng ^[2]; Saad, Yousef ^[1]

Univ. of Minnesota, Twin Cities, MN (United States)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

In this work, we describe a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure $$\mathcal{T}$$ based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSLR preconditioner follows a top-down traversal of $$\mathcal{T}$$ and exploits a low-rank property that is satisfied by the difference between the inverses of the local Schur complements and specific blocks of the original matrix. A few steps of the generalized Lanczos tridiagonalization procedure are applied to capture most of this difference. Numerical results are reported to illustrate the efficiency and robustness of the MSLR preconditioner with both two- and three-dimensional discretized PDE problems and with publicly available test problems.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1756735

Report Number(s):: LLNL-JRNL-680317; 804288

Journal Information:: SIAM Journal on Matrix Analysis and Applications, Vol. 37, Issue 1; ISSN 0895-4798

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

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