skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional based tight-binding

Journal Article · · Journal of Computational Chemistry
DOI:https://doi.org/10.1002/jcc.24254· OSTI ID:1248169
 [1];  [1];  [2];  [1];  [1]
  1. Argonne National Lab. (ANL), Argonne, IL (United States)
  2. Univ. of Alabama, Tuscaloosa, AL (United States)
+ Show Author Affiliations

The Shift-and-invert parallel spectral transformations (SIPs), a computational approach to solve sparse eigenvalue problems, is developed for massively parallel architectures with exceptional parallel scalability and robustness. The capabilities of SIPs are demonstrated by diagonalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall metallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studied is a 128,000 (2000) atom nanotube for which ~330,000 (~5600) eigenvalues and eigenfunctions are obtained in ~190 (~5) seconds when parallelized over 266,144 (16,384) Blue Gene/Q cores. Weak scaling and strong scaling of SIPs are analyzed and the performance of SIPs is compared with other novel methods. Different matrix ordering methods are investigated to reduce the cost of the factorization step, which dominates the time-to-solution at the strong scaling limit. As a result, a parallel implementation of assembling the density matrix from the distributed eigenvectors is demonstrated.

Research Organization:
Argonne National Laboratory (ANL), Argonne, IL (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences, and Biosciences Division
Grant/Contract Number:
AC02-06CH11357
OSTI ID:
1248169
Journal Information:
Journal of Computational Chemistry, Vol. 37, Issue 4; ISSN 0192-8651
Publisher:
WileyCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 13 works
Citation information provided by
Web of Science

References (61)

Simplified LCAO Method for the Periodic Potential Problem journal June 1954
Tight-binding models and density-functional theory journal June 1989
Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties journal September 1998
Tight-Binding Density Functional Theory:  An Approximate Kohn−Sham DFT Scheme † journal July 2007
Approximate Self‐Consistent Molecular Orbital Theory. I. Invariant Procedures journal November 1965
Ground states of molecules. 38. The MNDO method. Approximations and parameters journal June 1977
Optimization of parameters for semiempirical methods I. Method journal March 1989
Semiempirical quantum–chemical methods journal July 2013
Coupled-cluster theory in quantum chemistry journal February 2007
Inhomogeneous Electron Gas journal November 1964
Self-Consistent Equations Including Exchange and Correlation Effects journal November 1965
Assessment of Gaussian-3 and Density Functional Theories for Enthalpies of Formation of C 1 −C 16 Alkanes † journal June 2000
n -Alkane Isodesmic Reaction Energy Errors in Density Functional Theory Are Due to Electron Correlation Effects journal October 2010
Triplet states via intermediate neglect of differential overlap: Benzene, pyridine and the diazines journal January 1976
Semiempirical Molecular Orbital Methods book January 2007
ONIOM:  A Multilayered Integrated MO + MM Method for Geometry Optimizations and Single Point Energy Predictions. A Test for Diels−Alder Reactions and Pt(P( t -Bu) 3 ) 2 + H 2 Oxidative Addition journal January 1996
SIPs: Shift-and-invert parallel spectral transformations journal June 2007
Strategies for spectrum slicing based on restarted Lanczos methods journal March 2012
The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science journal May 2014
Parallel eigenvalue calculation based on multiple shift–invert Lanczos and contour integral based spectral projection method journal July 2014
New algorithms for iterative matrix-free eigensolvers in quantum chemistry journal December 2014
Density functional tight binding
  • Elstner, Marcus; Seifert, Gotthard
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 372, Issue 2011 https://doi.org/10.1098/rsta.2012.0483
journal March 2014
Direct calculation of electron density in density-functional theory journal March 1991
What is the best alternative to diagonalization of the Hamiltonian in large scale semiempirical calculations? journal January 1999
Linear scaling electronic structure methods journal July 1999
SIESTA-PEXSI: massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization journal July 2014
An iteration method for the solution of the eigenvalue problem of linear differential and integral operators journal October 1950
A Jacobi--Davidson Iteration Method for Linear Eigenvalue Problems journal January 2000
XIX. A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares journal August 1852
Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries book January 1997
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems journal September 2005
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling journal January 2001
Hybrid scheduling for the parallel solution of linear systems journal February 2006
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs journal January 1998
A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering journal January 1998
PT-Scotch: A tool for efficient parallel graph ordering journal July 2008
DFTB+, a Sparse Matrix-Based Implementation of the DFTB Method † journal July 2007
Computing the Minimum Fill-In is NP-Complete journal March 1981
Reducing the bandwidth of sparse symmetric matrices conference January 1969
Reading Althusser journal April 1971
Sparse Matrices in MATLAB: Design and Implementation journal January 1992
Nested Dissection of a Regular Finite Element Mesh journal April 1973
A Sparse Self-Consistent Field Algorithm and Its Parallel Implementation: Application to Density-Functional-Based Tight Binding journal May 2014
Parallel Computing in Quantum Chemistry reference-book January 2008
High-Performance Solvers for Dense Hermitian Eigenproblems journal January 2013
Elemental: A New Framework for Distributed Memory Dense Matrix Computations journal February 2013
Performance of Dense Eigensolvers on BlueGene/Q book January 2014
ScaLAPACK: a portable linear algebra library for distributed memory computers — design issues and performance journal August 1996
Fast semiempirical calculations: Fast Semiempirical Calculations journal June 1982
Calculations for millions of atoms with density functional theory: linear scaling shows its potential journal February 2010
Kohn−Sham Density Functional Theory Electronic Structure Calculations with Linearly Scaling Computational Time and Memory Usage journal December 2010
Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase journal March 2012
Sparse Projected-Gradient Method As a Linear-Scaling Low-Memory Alternative to Diagonalization in Self-Consistent Field Electronic Structure Calculations journal December 2012
Systematic sparse matrix error control for linear scaling electronic structure calculations journal January 2005
Numerical Methods for Electronic Structure Calculations of Materials journal January 2010
Pole-Based approximation of the Fermi-Dirac function journal August 2009
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix journal February 2011
A Fast Parallel Algorithm for Selected Inversion of Structured Sparse Matrices with Application to 2D Electronic Structure Calculations journal January 2011
Accelerating atomic orbital-based electronic structure calculation via pole expansion and selected inversion journal June 2013
A spectrum slicing method for the Kohn–Sham problem journal March 2012
Improving the scalability of a symmetric iterative eigensolver for multi-core platforms: IMPROVING THE SCALABILITY OF A SYMMETRIC ITERATIVE EIGENSOLVER journal September 2013

Cited By (5)

Three pillars for achieving quantum mechanical molecular dynamics simulations of huge systems: Divide-and-conquer, density-functional tight-binding, and massively parallel computation journal June 2016
Three-body expansion of the fragment molecular orbital method combined with density-functional tight-binding journal January 2017
SIESTA-SIPs: Massively parallel spectrum-slicing eigensolver for an ab initio molecular dynamics package journal August 2018
On the achievement of high fidelity and scalability for large-scale diagonalizations in grid-based DFT simulations journal March 2018
Recent advances in quantum‐mechanical molecular dynamics simulations of proton transfer mechanism in various water‐based environments journal May 2019

Similar Records

SIESTA-SIPs: Massively parallel spectrum-slicing eigensolver for an ab initio molecular dynamics package
Journal Article · Wed Aug 15 00:00:00 EDT 2018 · Journal of Computational Chemistry · OSTI ID:1248169
+5 more
A parallel implementation of tight-binding molecular dynamics based on reordering of atoms and the Lanczos eigen-solver
Book · Tue Dec 31 00:00:00 EST 1996 · OSTI ID:1248169
Toward an efficient parallel eigensolver for dense symmetric matrices
Journal Article · Fri Jan 01 00:00:00 EST 1999 · SIAM Journal on Scientific Computing · OSTI ID:1248169

Related Subjects

97 MATHEMATICS AND COMPUTING
36 MATERIALS SCIENCE
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY
sparse matrices
spectrum slicing
generalized eigenvalue problem
semi-empirical methods
tight-binding DFT