TY - JOUR

T1 - Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

AU - Auckenthaler, T.

AU - Blum, V.

AU - Bungartz, H. J.

AU - Huckle, T.

AU - Johanni, R.

AU - Krämer, L.

AU - Lang, B.

AU - Lederer, H.

AU - Willems, P. R.

N1 - Funding Information:
This work was supported by the Bundesministerium für Bildung und Forschung within the project “ELPA—Hochskalierbare Eigenwert-Löser für Petaflop-Großanwendungen”, Förderkennzeichen 01IH08007.

PY - 2011/12

Y1 - 2011/12

N2 - The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.

AB - The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.

KW - Blocked Householder transformations

KW - Divide-and-conquer tridiagonal eigensolver

KW - Eigenvalue and eigenvector computation

KW - Electronic structure calculations

KW - Parallelization

UR - http://www.scopus.com/inward/record.url?scp=81355138868&partnerID=8YFLogxK

U2 - 10.1016/j.parco.2011.05.002

DO - 10.1016/j.parco.2011.05.002

M3 - Article

AN - SCOPUS:81355138868

SN - 0167-8191

VL - 37

SP - 783

EP - 794

JO - Parallel Computing

JF - Parallel Computing

IS - 12

ER -