Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations

Abstract

The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials' simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale two-dimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. Employing 55 296 computational cores,more » the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.« less

Authors:
 [1];  [2]; ORCiD logo [3];  [3];  [4]
+ Show Author Affiliations
  1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
  2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Divisio; Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
  3. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Divisio
  4. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Physics Division
Publication Date:
Research Org.:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1438776
Alternate Identifier(s):
OSTI ID: 1420728; OSTI ID: 1456961
Report Number(s):
LLNL-JRNL-735618
Journal ID: ISSN 0021-9606; TRN: US1900523
Grant/Contract Number:  
AC52-07NA27344; AC52- 07NA27344; AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 145; Journal Issue: 15; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

Citation Formats

Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, and Pask, John E. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations. United States: N. p., 2016. Web. doi:10.1063/1.4964861.
Copy to clipboard
Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, & Pask, John E. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations. United States. https://doi.org/10.1063/1.4964861
Copy to clipboard
Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, and Pask, John E. Fri . "Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations". United States. https://doi.org/10.1063/1.4964861. https://www.osti.gov/servlets/purl/1438776.
Copy to clipboard
@article{osti_1438776,
title = {Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations},
author = {Banerjee, Amartya S. and Lin, Lin and Hu, Wei and Yang, Chao and Pask, John E.},
abstractNote = {The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials' simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale two-dimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. Employing 55 296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.},
doi = {10.1063/1.4964861},
journal = {Journal of Chemical Physics},
number = 15,
volume = 145,
place = {United States},
year = {Fri Oct 21 00:00:00 EDT 2016},
month = {Fri Oct 21 00:00:00 EDT 2016}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1063/1.4964861
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Citation Metrics:
Cited by: 26 works
Citation information provided by
Web of Science

Figures / Tables:

FIG. 1 FIG. 1: Partitioning of a domain into DG elements and the resulting discretized Hamiltonian HDG.
All figures and tables (11 total)
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Large-scale ab initio calculations based on three levels of parallelization
journal, April 2008

RESCU: A real space electronic structure method
journal, February 2016

Property-optimized Gaussian basis sets for molecular response calculations
journal, October 2010

  • Rappoport, Dmitrij; Furche, Filipp
  • The Journal of Chemical Physics, Vol. 133, Issue 13
  • DOI: 10.1063/1.3484283

Reorthogonalization and Stable Algorithms for Updating the Gram-Schmidt QR Factorization
journal, October 1976

  • Daniel, J. W.; Gragg, W. B.; Kaufman, L.
  • Mathematics of Computation, Vol. 30, Issue 136
  • DOI: 10.2307/2005398

A spectral scheme for Kohn–Sham density functional theory of clusters
journal, April 2015

  • Banerjee, Amartya S.; Elliott, Ryan S.; James, Richard D.
  • Journal of Computational Physics, Vol. 287
  • DOI: 10.1016/j.jcp.2015.02.009

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
journal, December 2013

Architecture of Qbox: A scalable first-principles molecular dynamics code
journal, January 2008

  • Gygi, F.
  • IBM Journal of Research and Development, Vol. 52, Issue 1.2
  • DOI: 10.1147/rd.521.0137

Linear-scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP
journal, July 2009

  • Hine, N. D. M.; Haynes, P. D.; Mostofi, A. A.
  • Computer Physics Communications, Vol. 180, Issue 7
  • DOI: 10.1016/j.cpc.2008.12.023

Adaptive Finite Element Approximations for Kohn--Sham Models
journal, January 2014

  • Chen, Huajie; Dai, Xiaoying; Gong, Xingao
  • Multiscale Modeling & Simulation, Vol. 12, Issue 4
  • DOI: 10.1137/130916096

Finite-difference-pseudopotential method: Electronic structure calculations without a basis
journal, February 1994

octopus: a tool for the application of time-dependent density functional theory
journal, September 2006

  • Castro, Alberto; Appel, Heiko; Oliveira, Micael
  • physica status solidi (b), Vol. 243, Issue 11
  • DOI: 10.1002/pssb.200642067

\mathcal{O}(N) methods in electronic structure calculations
journal, February 2012

Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory
journal, January 2015

  • Hu, Wei; Lin, Lin; Yang, Chao
  • Physical Chemistry Chemical Physics, Vol. 17, Issue 47
  • DOI: 10.1039/C5CP00333D

Efficacious Form for Model Pseudopotentials
journal, May 1982

Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme
journal, May 1957

  • Bauer, Friedrich L.
  • Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 8, Issue 3
  • DOI: 10.1007/BF01600502

Large-Scale First-Principles Molecular Dynamics simulations on the BlueGene/L Platform using the Qbox code
conference, January 2005

  • Gygi, F.; Yates, R. K.; Lorenz, J.
  • ACM/IEEE SC 2005 Conference (SC'05)
  • DOI: 10.1109/SC.2005.40

Electronic Structure
book, January 2004

A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix
journal, June 2015

Linear-scaling first-principles molecular dynamics with plane-waves accuracy
journal, March 2006

Reliability of layered neural oscillator networks
journal, January 2009

  • Lin, Kevin. K.; Shea-Brown, Eric; Young, Lai-Sang
  • Communications in Mathematical Sciences, Vol. 7, Issue 1
  • DOI: 10.4310/CMS.2009.v7.n1.a12

Self-Consistent Equations Including Exchange and Correlation Effects
journal, November 1965

Parallel eigensolvers in plane-wave Density Functional Theory
journal, February 2015

Convergence acceleration of iterative sequences. the case of scf iteration
journal, July 1980

Towards Very Large-Scale Electronic-Structure Calculations
journal, February 1992

Higher-order finite-difference pseudopotential method: An application to diatomic molecules
journal, October 1994

ScaLAPACK Users' Guide
book, January 1997

  • Blackford, L. S.; Choi, J.; Cleary, A.
  • Society for Industrial and Applied Mathematics
  • DOI: 10.1137/1.9780898719642

SIESTA-PEXSI: massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization
journal, July 2014

Non-periodic finite-element formulation of Kohn–Sham density functional theory
journal, February 2010

  • Suryanarayana, Phanish; Gavini, Vikram; Blesgen, Thomas
  • Journal of the Mechanics and Physics of Solids, Vol. 58, Issue 2
  • DOI: 10.1016/j.jmps.2009.10.002

Accelerating atomic orbital-based electronic structure calculation via pole expansion and selected inversion
journal, June 2013

Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals
journal, September 1969

  • Hehre, W. J.; Stewart, R. F.; Pople, J. A.
  • The Journal of Chemical Physics, Vol. 51, Issue 6
  • DOI: 10.1063/1.1672392

DGDFT: A massively parallel method for large scale density functional theory calculations
journal, September 2015

  • Hu, Wei; Lin, Lin; Yang, Chao
  • The Journal of Chemical Physics, Vol. 143, Issue 12
  • DOI: 10.1063/1.4931732

Electronic-structure calculations based on the finite-element method
journal, August 1995

<i>A posteriori</i> error estimator for adaptive local basis functions to solve Kohn–Sham density functional theory
journal, January 2015

Simultaneous iteration method for symmetric matrices
journal, December 1970

Simplified LCAO Method for the Periodic Potential Problem
journal, June 1954

Self-consistent-field calculations using Chebyshev-filtered subspace iteration
journal, November 2006

  • Zhou, Yunkai; Saad, Yousef; Tiago, Murilo L.
  • Journal of Computational Physics, Vol. 219, Issue 1
  • DOI: 10.1016/j.jcp.2006.03.017

The SIESTA method for ab initio order- N materials simulation
journal, March 2002

  • Soler, José M.; Artacho, Emilio; Gale, Julian D.
  • Journal of Physics: Condensed Matter, Vol. 14, Issue 11
  • DOI: 10.1088/0953-8984/14/11/302

Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
journal, October 1996

Linear scaling electronic structure methods
journal, July 1999

Super-matrix methods
journal, May 1989

Improved accuracy and acceleration of variational order- N electronic-structure computations by projection techniques
journal, November 1998

Cyclic density functional theory: A route to the first principles simulation of bending in nanostructures
journal, November 2016

  • Banerjee, Amartya S.; Suryanarayana, Phanish
  • Journal of the Mechanics and Physics of Solids, Vol. 96
  • DOI: 10.1016/j.jmps.2016.08.007

Inhomogeneous Electron Gas
journal, November 1964

Higher-order adaptive finite-element methods for Kohn–Sham density functional theory
journal, November 2013

Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration
journal, December 2006

Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn–Sham equation
journal, October 2014

  • Zhou, Yunkai; Chelikowsky, James R.; Saad, Yousef
  • Journal of Computational Physics, Vol. 274
  • DOI: 10.1016/j.jcp.2014.06.056

A spectrum slicing method for the Kohn–Sham problem
journal, March 2012

  • Schofield, Grady; Chelikowsky, James R.; Saad, Yousef
  • Computer Physics Communications, Vol. 183, Issue 3
  • DOI: 10.1016/j.cpc.2011.11.005

Ab initio molecular simulations with numeric atom-centered orbitals
journal, November 2009

  • Blum, Volker; Gehrke, Ralf; Hanke, Felix
  • Computer Physics Communications, Vol. 180, Issue 11
  • DOI: 10.1016/j.cpc.2009.06.022

Finite element methods in ab initio electronic structure calculations
journal, April 2005

  • Pask, J. E.; Sterne, P. A.
  • Modelling and Simulation in Materials Science and Engineering, Vol. 13, Issue 3
  • DOI: 10.1088/0965-0393/13/3/R01

Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase
journal, March 2012

  • VandeVondele, Joost; Borštnik, Urban; Hutter, Jürg
  • Journal of Chemical Theory and Computation, Vol. 8, Issue 10
  • DOI: 10.1021/ct200897x

Density-matrix-based algorithm for solving eigenvalue problems
journal, March 2009

Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients
journal, October 1992

An Interior Penalty Finite Element Method with Discontinuous Elements
journal, August 1982

  • Arnold, Douglas N.
  • SIAM Journal on Numerical Analysis, Vol. 19, Issue 4
  • DOI: 10.1137/0719052

Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations
journal, March 2016

Computational aspects of F. L. Bauer's simultaneous iteration method
journal, March 1969

The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices
journal, January 1975

Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation
journal, February 2012

Linear-scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP
journal, July 2009

  • Hine, N. D. M.; Haynes, P. D.; Mostofi, A. A.
  • Computer Physics Communications, Vol. 180, Issue 7
  • DOI: 10.1016/j.cpc.2008.12.023
    Sort by:
    [ × clear filter / sort ]

    Works referencing / citing this record:

    Ab initio simulations of liquid electrolytes for energy conversion and storage
    journal, October 2018

    • Pham, Tuan Anh
    • International Journal of Quantum Chemistry, Vol. 119, Issue 1
    • DOI: 10.1002/qua.25795

    Orbital-enriched flat-top partition of unity method for the Schrödinger eigenproblem
    journal, December 2018

    • Albrecht, Clelia; Klaar, Constanze; Pask, John Ernest
    • Computer Methods in Applied Mechanics and Engineering, Vol. 342
    • DOI: 10.1016/j.cma.2018.07.042

    ELSI — An open infrastructure for electronic structure solvers
    journal, November 2020

    • Yu, Victor Wen-zhe; Campos, Carmen; Dawson, William
    • Computer Physics Communications, Vol. 256
    • DOI: 10.1016/j.cpc.2020.107459

    Task-based Parallel Computation of the Density Matrix in Quantum-based Molecular Dynamics using Graph Partitioning
    journal, January 2017

    • Ghale, Purnima; Kroonblawd, Matthew P.; Mniszewski, Sue
    • SIAM Journal on Scientific Computing, Vol. 39, Issue 6
    • DOI: 10.1137/16m109404x

    ChASE: Chebyshev Accelerated Subspace iteration Eigensolver for sequences of Hermitian eigenvalue problems
    preprint, January 2018

    Discontinuous Galerkin discretization for quantum simulation of chemistry
    preprint, January 2019

      Sort by:
      [ × clear filter / sort ]

      Figures / Tables found in this record:

      FIG. 1 (p. 12)figure
      Algorithm 1 (p. 15)figure
      FIG. 2 (p. 16)figure
      FIG. 3 (p. 21)figure
      FIG. 4 (p. 22)figure
      FIG. 5 (p. 25)figure
      FIG. 6 (p. 26)figure
      FIG. 7 (p. 28)figure
      FIG. 8 (p. 29)figure
      TABLE I (p. 30)table
      TABLE II (p. 31)table
        [ × clear filter / sort ]
        Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.

        Similar Records in DOE PAGES and OSTI.GOV collections: