A parallel preconditioner is proposed for general large sparse linear systems that combines a power series expansion method with low-rank correction techniques. To enhance convergence, a power series expansion is added to a basic Schur complement iterative scheme by exploiting a standard matrix splitting of the Schur complement. One of the goals of the power series approach is to improve the eigenvalue separation of the preconditioner thus allowing an effective application of a low-rank correction technique. Experiments indicate that this combination can be quite robust when solving highly indefinite linear systems. The preconditioner exploits a domain-decomposition approach, and its construction starts with the use of a graph partitioner to reorder the original coefficient matrix. In this framework, unknowns corresponding to interface variables are obtained by solving a linear system whose coefficient matrix is the Schur complement. Unknowns associated with the interior variables are obtained by solving a block diagonal linear system where parallelism can be easily exploited. Numerical examples are provided to illustrate the effectiveness of the proposed preconditioner, with an emphasis on highlighting its robustness properties in the indefinite case.
Bibliographical noteFunding Information:
\ast Received by the editors January 31, 2020; accepted for publication (in revised form) by L. Giraud January 5, 2021; published electronically April 27, 2021. https://doi.org/10.1137/20M1316445 Funding: The work of the first author was supported by the National Natural Science Foundation of China grant 12001311 and the Shuimu Scholar of Tsinghua University. The work of the second author was supported by the National Science Foundation (NSF) grant OAC-2003720. The work of the third author was supported by the NSF grant DMS-1912048 and the Minnesota Supercomputing Institute. \dagger Department of Mathematical Sciences, Tsinghua University, Bejing, China (zheng1990@mail. tsinghua.edu.cn). \ddagger Department of Mathematics, Emory University, Atlanta, GA 30322 USA (email@example.com). \S Computer Science \& Engineering, University of Minnesota, Twin Cities, Minneapolis, MN 55455-0154 USA (firstname.lastname@example.org).
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- Domain decomposition
- Krylov subspace method
- Low-rank correction
- Parallel preconditioner
- Power series expansion
- Schur complement