Beyond AMLS: Domain decomposition with rational filtering

TL;DR

This paper introduces a purely algebraic rational filtering domain decomposition method for large sparse symmetric eigenvalue problems, enabling parallel computation and improved efficiency without eigenvalue count estimation.

Contribution

It presents a novel algebraic domain decomposition technique that integrates partial matrix resolvent filtering with interface problem solving, enhancing rational filtering methods.

Findings

Competitive performance against rational filtering Krylov methods.

Parallelizable solution of interior subproblems.

No need for eigenvalue count estimation.

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 lying 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.