## Abstract

This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem associated with each subdomain into two disjoint subproblems. The first subproblem is associated with the interface variables and accounts for the interaction among neighboring subdomains. To compute the solution of the original eigenvalue problem at the interface variables we leverage ideas from contour integral eigenvalue solvers. The second subproblem is associated with the interior variables in each subdomain and can be solved in parallel among the different subdomains using real arithmetic only. Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary to vectors whose length is equal to the number of interface variables. In addition, no estimation of the number of eigenvalues located inside the interval of interest is needed. Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches.

Original language | English (US) |
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Pages (from-to) | C477-C502 |

Journal | SIAM Journal on Scientific Computing |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Funding Information:∗Submitted to the journal’s Software and High-Performance Computing section October 30, 2017; accepted for publication (in revised form) April 19, 2018; published electronically July 12, 2018. http://www.siam.org/journals/sisc/40-4/M115452.html Funding: This work was supported by NSF (award CCF-1505970). The work of the first author was partially supported by a Gerondelis Foundation Fellowship. †Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 (kalan019@umn.edu, saad@umn.edu). ‡Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322 (yxi26@emory.edu).

Publisher Copyright:

© 2018 Society for Industrial and Applied Mathematics.

## Keywords

- Domain decomposition
- Parallel computing
- Rational filtering
- Schur complement
- Symmetric generalized eigenvalue problem