Robust algebraic Schur complement preconditioners based on low rank corrections

Laura Grigori 1 Frédéric Nataf 1, 2 Soleiman Yousef 1
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, Institut National des Sciences Mathématiques et de leurs Interactions
Abstract : In this paper we introduce LORASC, a robust algebraic preconditioner for solving sparse linear systems of equations involving symmetric and positive definite matrices. The graph of the input matrix is partitioned by using k-way partitioning with vertex separators into N disjoint domains and a separator formed by the vertices connecting the N domains. The obtained permuted matrix has a block arrow structure. The preconditioner relies on the Cholesky factorization of the first N diagonal blocks and on approximating the Schur complement corresponding to the separator block. The approximation of the Schur complement involves the factorization of the last diagonal block and a low rank correction obtained by solving a generalized eigenvalue problem or a randomized algorithm. The preconditioner can be build and applied in parallel. Numerical results on a set of matrices arising from the discretization by the finite element method of linear elasticity models illustrate the robusteness and the efficiency of our preconditioner.
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Rapport
[Research Report] RR-8557, INRIA. 2014, pp.18
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Laura Grigori, Frédéric Nataf, Soleiman Yousef. Robust algebraic Schur complement preconditioners based on low rank corrections. [Research Report] RR-8557, INRIA. 2014, pp.18. 〈hal-01017448〉

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