A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

Abstract

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidsontype methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

General build up of basis and matrix in the diagonalization approach. Determination of Kramers configuration state functions

“…For comparison purposes, sparse matrices approaches have been also accounted for due to the nature of the matrix representation. From the relative large range of possible solvers and implementations, the SLEPc suite has been chosen and the Krylov‐Shur procedure has been chosen as the default one, although the Jacobi‐Davidson procedure as implemented in SLEPc has been also tested for lower number of roots requested. Optional packages which can be used with SLEPc such as ARPACK, PRIMME, BLZPACK, TRLAN, BLOPEX, and/or FEAST have not been installed with the compilation of SLEPc and the required PETSc interface.…”

“…This includes possible extensions to a larger number of open shells with targeting a given KCSFs function using Full Configuration Interaction (FCI) diagonalization strategies,. For example, Davidson and/or alternatives aside of the employed Krylov‐Shur sparse matrices treatment, or the comparison to an in core Jacobi‐Davidson procedure . Furthermore, the usage of KCSFs in a general case of N fermions in M shells will be mentioned.…”

Parallel finite element density functional computations exploiting grid refinement and subspace recycling

“…These computations are carried out iteratively, where the extracted approximations are used to improve the subspace and extract better approximations, until the computed approximations are sufficiently accurate. For background material on projection methods, the reader is referred to [20,21] and references therein.…”

GPU-Accelerated LOBPCG Method with Inexact Null-Space Filtering for Solving Generalized Eigenvalue Problems in Computational Electromagnetics Analysis with Higher-Order FEM

“…As the size of the matrix and the number of nonzero matrix elements grow, the time required to solve such sparse generalized eigenproblems increases. To mitigate this problem, parallel computing techniques have been proposed [8][9][10]. In most cases massively parallel implementations are intended for clusters, but such computer systems are costly and not readily available for researchers and engineers.…”