PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

“…This will be illustrated by comparing the FEAST algorithm implemented with a domain decomposition-based direct solver with an implementation of FEAST that uses a standard parallel sparse direct solver. Similar behavior was observed in the work of Kestyn et al 14 for using an application-specific domain decomposition linear system solver in FEAST. In contrast, in this paper, we consider domain decomposition using algebraic partitionings obtained by a graph partitioner.…”

“…On the other hand, increasing N c and distributing the quadrature nodes, as in CI‐M2, seem to be a more efficient choice when a larger number of computational resources become available. Additional details on the performance of FEAST and related approaches when multiple levels of MPI parallelism are considered can be found in other works …”

“…The accuracy of the approximate eigenpairs computed by a Rayleigh–Ritz projection on the subspace created by can be improved by repeating the procedure in , using the most recent approximate eigenvectors as the new matrix V to multiply . If direct solvers are used to solve the complex linear systems in , this approach essentially amounts to subspace iteration with the matrix A replaced by ρ ( A ), that is, the FEAST package ; see also other works . It is also possible to consider contour integrals of other rational functions, for example, the scalar function u ∗ ( ζ I − A ) −1 v , with , as proposed in other works .…”

“…Assuming that V ∈ R n×r is chosen such that V T X has rank r, (14) captures the exact invariant subspace of A associated with the sought eigenvalues, and Z ≡ V can be exploited in a Rayleigh-Ritz projection to recover the actual eigenpairs of A. Because the scheme discussed above considers all blocks of , we will refer to it as "domain decomposition full projector" (DD-FP).…”

“…In practice, the contour integrals in (14) will have to be approximated numerically. Once a quadrature rule is selected, with quadrature nodes and weights { j , j }, j = 1, … , 2N c , (14) is approximated by the following summations:…”