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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]
  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 Laboratory
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC-23)
OSTI Identifier:
1395271
Report Number(s):
PNNL-SA-114405
Journal ID: ISSN 0010-4655; PII: S0010465517302370
Grant/Contract Number:
AC05-76RL01830; AC02-05CH1123; AC02-05CH11231; KC-030106062653
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Computer Physics Communications
Additional Journal Information:
Journal Volume: 221; 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.
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, & Yang, Chao. Efficient block preconditioned eigensolvers for linear response time-dependent 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 time-dependent density functional theory". United States. doi:10.1016/J.CPC.2017.07.017.
@article{osti_1395271,
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},
journal = {Computer Physics Communications},
number = ,
volume = 221,
place = {United States},
year = 2017,
month = 8
}

Journal Article:
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  • 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 self-adjoint with respect to a K-inner 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 K-innermore » 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.« less
  • We discuss our implementation and application of time-dependent density functional theory (TDDFT) to core-level near-edge absorption spectroscopy, using both linear-response (LR) and real-time (RT) approaches. We briefly describe our restricted window TDDFT (REWTDDFT) approach for core excitations which has also been reported by others groups. This is followed by a detailed discussion of real-time TDDFT techniques tailored to core excitations, including obtaining spectral information through delta-function excitation, post-processing time-dependent signals, and resonant excitation through quasi-monochromatic excitation. We present results for the oxygen K-edge of water and carbon dioxide; the carbon K-edge of carbon dioxide; the ruthenium L3-edge for the hexaamminerutheium(III)more » ion, including scalar relativistic corrections via the zeroth order regular approximation (ZORA); and the carbon and fluorine K-edges for a series of fluorobenzenes. In all cases, the calculated spectra are found to be in good agreement with experiment, requiring only a uniform shift on the order of a few percent. Real-time TDDFT visualization of excited state charge densities are used to visually examine the nature of each excitation, which gives insight into the effects of atoms bound to the absorbing center.« less
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