Title: A sparse matrix–vector multiplication based algorithm for accurate density matrix computations on systems of millions of atoms

Abstract

Here we present an efficient sparse matrix–vector (SpMV) based method to compute the density matrix P from a given Hamiltonian in electronic structure computations. Our method is a hybrid approach based on Chebyshev–Jackson approximation theory and matrix purification methods like the second order spectral projection purification (SP2). Recent methods to compute the density matrix scale as O(N) in the number of floating point operations but are accompanied by large memory and communication overhead, and they are based on iterative use of the sparse matrix–matrix multiplication kernel (SpGEMM), which is known to be computationally irregular. In addition to irregularity in the sparse Hamiltonian H, the nonzero structure of intermediate estimates of P depends on products of H and evolves over the course of computation. On the other hand, an expansion of the density matrix P in terms of Chebyshev polynomials is straightforward and SpMV based; however, the resulting density matrix may not satisfy the required constraints exactly. In this paper, we analyze the strengths and weaknesses of the Chebyshev–Jackson polynomials and the second order spectral projection purification (SP2) method, and propose to combine them so that the accurate density matrix can be computed using the SpMV computational kernel only, and withoutmore » having to store the density matrix P. Our method accomplishes these objectives by using the Chebyshev polynomial estimate as the initial guess for SP2, which is followed by using sparse matrix–vector multiplications (SpMVs) to replicate the behavior of the SP2 algorithm for purification. We demonstrate the method on a tight-binding model system of an oxide material containing more than 3 million atoms. In addition, we also present the predicted behavior of our method when applied to near-metallic Hamiltonians with a wide energy spectrum.« less

Authors:

Ghale, Purnima  ^[1];

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Johnson, Harley T. ^[1]

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1. University of Illinois at Urbana-Champaign, IL (United States)

Publication Date:

Thu Feb 15 00:00:00 EST 2018

Research Org.:

Univ. of Illinois at Urbana-Champaign, IL (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1538209

Alternate Identifier(s):

OSTI ID: 1547122

Grant/Contract Number:  

NA0002374

Resource Type:

Accepted Manuscript

Journal Name:

Computer Physics Communications

Additional Journal Information:

Journal Volume: 227; Journal Issue: C; Journal ID: ISSN 0010-4655

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

42 ENGINEERING; matrix purification; kernel polynomial; density matrix