A hierarchical low rank schur complement preconditioner for indefinite linear systems

Geoffrey Dillon; Vassilis Kalantzis; Yuanzhe Xi; Yousef Saad

doi:10.1137/17M1143320

A hierarchical low rank schur complement preconditioner for indefinite linear systems

Geoffrey Dillon, Vassilis Kalantzis, Yuanzhe Xi, Yousef Saad

Computer Science and Engineering

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods. This paper combines ideas from the multilevel Schur low rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235–259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure T that represents a hierarchical decomposition of the original matrix. This decomposition gives rise to a block structured matrix at each level of T . An approximate inverse of the original matrix based on its block LU factorization is computed at each level via a low rank property that characterizes the difference between the inverses of the Schur complement and another block of the reordered matrix. The low rank correction matrix is computed by several steps of the Arnoldi process. Numerical results illustrate the robustness of the proposed preconditioner with respect to indefiniteness for a few discretized partial differential equations and publicly available test problems.

Original language	English (US)
Pages (from-to)	A2234-A2252
Journal	SIAM Journal on Scientific Computing
Volume	40
Issue number	4
DOIs	https://doi.org/10.1137/17M1143320
State	Published - 2018

Bibliographical note

Funding Information:
This work was supported by NSF under grant DMS-1521573 and by the Minnesota Supercomputing Institute. The authors would like to thank the Minnesota Supercomputing Institute for the use of their extensive computing resources and the anonymous referees for their careful reading of this paper and helpful suggestions.

Funding Information:
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section August 14, 2017; accepted for publication (in revised form) April 3, 2018; published electronically July 17, 2018. http://www.siam.org/journals/sisc/40-4/M114332.html Funding: This work was supported by NSF under grant DMS-1521573 and by the Minnesota Supercomputing Institute. †Department of Computer Science & Engineering, University of Minnesota, Twin Cities, Minneapolis, MN 55455 (gdillon@umn.edu, kalan019@umn.edu, yxi@umn.edu, saad@umn.edu).

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

Keywords

Block preconditioner
Domain decomposition
Krylov subspace methods
Low rank approximation
Multilevel
Nested dissection ordering
Schur complements

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title = "A hierarchical low rank schur complement preconditioner for indefinite linear systems",

abstract = "Nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods. This paper combines ideas from the multilevel Schur low rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235–259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure T that represents a hierarchical decomposition of the original matrix. This decomposition gives rise to a block structured matrix at each level of T . An approximate inverse of the original matrix based on its block LU factorization is computed at each level via a low rank property that characterizes the difference between the inverses of the Schur complement and another block of the reordered matrix. The low rank correction matrix is computed by several steps of the Arnoldi process. Numerical results illustrate the robustness of the proposed preconditioner with respect to indefiniteness for a few discretized partial differential equations and publicly available test problems.",

keywords = "Block preconditioner, Domain decomposition, Krylov subspace methods, Low rank approximation, Multilevel, Nested dissection ordering, Schur complements",

author = "Geoffrey Dillon and Vassilis Kalantzis and Yuanzhe Xi and Yousef Saad",

note = "Funding Information: This work was supported by NSF under grant DMS-1521573 and by the Minnesota Supercomputing Institute. The authors would like to thank the Minnesota Supercomputing Institute for the use of their extensive computing resources and the anonymous referees for their careful reading of this paper and helpful suggestions. Funding Information: ∗Submitted to the journal{\textquoteright}s Methods and Algorithms for Scientific Computing section August 14, 2017; accepted for publication (in revised form) April 3, 2018; published electronically July 17, 2018. http://www.siam.org/journals/sisc/40-4/M114332.html Funding: This work was supported by NSF under grant DMS-1521573 and by the Minnesota Supercomputing Institute. †Department of Computer Science & Engineering, University of Minnesota, Twin Cities, Minneapolis, MN 55455 (gdillon@umn.edu, kalan019@umn.edu, yxi@umn.edu, saad@umn.edu). Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics.",

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N2 - Nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods. This paper combines ideas from the multilevel Schur low rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235–259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure T that represents a hierarchical decomposition of the original matrix. This decomposition gives rise to a block structured matrix at each level of T . An approximate inverse of the original matrix based on its block LU factorization is computed at each level via a low rank property that characterizes the difference between the inverses of the Schur complement and another block of the reordered matrix. The low rank correction matrix is computed by several steps of the Arnoldi process. Numerical results illustrate the robustness of the proposed preconditioner with respect to indefiniteness for a few discretized partial differential equations and publicly available test problems.

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A hierarchical low rank schur complement preconditioner for indefinite linear systems

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