Efficient block preconditioned eigensolvers for linear response timedependent density functional theory
Abstract
We present two efficient iterative algorithms for solving the linear response eigen value 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 a product eigenvalue problem that is selfadjoint with respect to a Kinner product. 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 Kinner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.
 Authors:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Lab.
 Publication Date:
 Research Org.:
 Lawrence Berkeley National Laboratory (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) (SC22); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21); Ministry of Education, Youth and Sports (Cambodia)
 OSTI Identifier:
 1406765
 Report Number(s):
 PNNLSA114405
Journal ID: ISSN 00104655; 48614; KC0301060
 DOE Contract Number:
 AC0205CH11231; AC0205CH1123; KC030106062653; AC0576RL1830
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Computer Physics Communications; Journal Volume: 221; Journal Issue: C
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Eigensolvers; Linear Response Timedependent Density Functional Theory; Environmental Molecular Sciences Laboratory
Citation Formats
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. Efficient block preconditioned eigensolvers for linear response timedependent density functional theory. United States: N. p., 2017.
Web. doi:10.1016/j.cpc.2017.07.017.
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, & Yang, Chao. Efficient block preconditioned eigensolvers for linear response timedependent density functional theory. United States. doi:10.1016/j.cpc.2017.07.017.
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. 2017.
"Efficient block preconditioned eigensolvers for linear response timedependent density functional theory". United States.
doi:10.1016/j.cpc.2017.07.017.
@article{osti_1406765,
title = {Efficient block preconditioned eigensolvers for linear response timedependent density functional theory},
author = {Vecharynski, Eugene and Brabec, Jiri and Shao, Meiyue and Govind, Niranjan and Yang, Chao},
abstractNote = {We present two efficient iterative algorithms for solving the linear response eigen value 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 a product eigenvalue problem that is selfadjoint with respect to a Kinner product. 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 Kinner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.},
doi = {10.1016/j.cpc.2017.07.017},
journal = {Computer Physics Communications},
number = C,
volume = 221,
place = {United States},
year = 2017,
month =
}

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 selfadjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore »Cited by 1

Efficient block preconditioned eigensolvers for linear response timedependent density functional theory
In this article, 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 selfadjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore »Cited by 1 
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