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Title: Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Abstract

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-artmore » Davidson type solvers by a factor of two in both solution time and storage.« less

Authors:
 [1];  [1];  [1];  [2];  [1]
+ Show Author Affiliations
  1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
  2. Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Lab.
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Ministry of Education, Youth and Sports (Cambodia); USDOE Office of Science (SC), Biological and Environmental Research (BER)
OSTI Identifier:
1425428
Alternate Identifier(s):
OSTI ID: 1395271; OSTI ID: 1576835
Report Number(s):
PNNL-SA-114405
Journal ID: ISSN 0010-4655; ark:/13030/qt8rf6x4kg; TRN: US1802105
Grant/Contract Number:  
AC02-05CH11231; AC02-05CH1123; KC-030106062653; AC05-76RL01830; AC05-76RL1830
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Computer Physics Communications
Additional Journal Information:
Journal Volume: 221; Journal Issue: C; Journal ID: ISSN 0010-4655
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Time dependent density functional theory; Linear response eigenvalue problem; Preconditioned eigensolvers

Citation Formats

Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory. United States: N. p., 2017. Web. doi:10.1016/j.cpc.2017.07.017.
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Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, & Yang, Chao. Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory. United States. https://doi.org/10.1016/j.cpc.2017.07.017
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Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. 2017. "Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory". United States. https://doi.org/10.1016/j.cpc.2017.07.017. https://www.osti.gov/servlets/purl/1425428.
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@article{osti_1425428,
title = {Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory},
author = {Vecharynski, Eugene and Brabec, Jiri and Shao, Meiyue and Govind, Niranjan and Yang, Chao},
abstractNote = {Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.},
doi = {10.1016/j.cpc.2017.07.017},
url = {https://www.osti.gov/biblio/1425428}, journal = {Computer Physics Communications},
issn = {0010-4655},
number = C,
volume = 221,
place = {United States},
year = {Fri Dec 01 00:00:00 EST 2017},
month = {Fri Dec 01 00:00:00 EST 2017}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1016/j.cpc.2017.07.017
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Cited by: 8 works
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Figures / Tables:

Algorithm 1 Algorithm 1: The Davidson algorithm8 for solving the linear response eigenvalue problem.
All figures and tables (10 total)
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    S erenity : A subsystem quantum chemistry program
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    A FEAST Algorithm for the Linear Response Eigenvalue Problem
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      Figures / Tables found in this record:

      Algorithm 1(p. 9)figure
      Algorithm 2(p. 11)figure
      Algorithm 3(p. 15)figure
      Algorithm 4(p. 17)figure
      Algorithm 5(p. 21)figure
      Algorithm 6(p. 22)figure
      Figure 1(p. 23)figure
      Figure 2(p. 25)figure
      Figure 3(p. 26)figure
      Figure 4(p. 27)figure
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