A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]

Vecharynski, Eugene; Yang, Chao; Pask, John E.

doi:10.1016/j.jcp.2015.02.030

Title: A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]

Journal Article · Wed Feb 25 00:00:00 EST 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2015.02.030· OSTI ID:1325863

Vecharynski, Eugene ^[1]; Yang, Chao ^[1]; Pask, John E. ^[2]

Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Here, we present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of A. These problems arise from, for example, density functional theory (DFT) based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh–Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: AC52-07NA27344; AC02-05CH11231

OSTI ID:: 1325863

Alternate ID(s):: OSTI ID: 1556219

Report Number(s):: LLNL-JRNL-695289

Journal Information:: Journal of Computational Physics, Vol. 290, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 26 works

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Prediction of atomization energy using graph kernel and active learning Tang, Yu-Hang; de Jong, Wibe A. The Journal of Chemical Physics, Vol. 150, Issue 4 https://doi.org/10.1063/1.5078640	journal	January 2019
Convergence theory for preconditioned eigenvalue solvers in a nutshell Argentati, Merico E.; Knyazev, Andrew V.; Neymeyr, Klaus arXiv https://doi.org/10.48550/arxiv.1412.5005	text	January 2014

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preconditioned eigenvalue solvers
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Title: A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]

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