Schur complement-based domain decomposition preconditioners with low-rank corrections

Ruipeng Li, Yuanzhe Xi, Yousef Saad

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


This paper introduces a robust preconditioner for general sparse matrices based on low-rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low-rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach.

Original languageEnglish (US)
Pages (from-to)706-729
Number of pages24
JournalNumerical Linear Algebra with Applications
Issue number4
StatePublished - Aug 1 2016


  • Krylov subspace method
  • domain decomposition
  • general sparse linear system
  • low-rank approximation
  • parallel preconditioner
  • the Lanczos algorithm


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