SQDFT: Spectral Quadrature method for large-scale parallel O(N) Kohn–Sham calculations at high temperature

Abstract

We present SQDFT: a large-scale parallel implementation of the Spectral Quadrature (SQ) method for O(N) Kohn-Sham Density Functional Theory (DFT) calculations at high temperature. Specifically, we develop an efficient and scalable finite-difference implementation of the infinite-cell Clenshaw-Curtis SQ approach, in which results for the infinite crystal are obtained by expressing quantities of interest as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw-Curtis quadrature rules. We demonstrate the accuracy of SQDFT by showing systematic convergence of energies and atomic forces with respect to SQ parameters to reference diagonalization results, and convergence with discretization to established planewave results, for both metallic and insulating systems. We further demonstrate that SQDFT achieves excellent strong and weak parallel scaling on computer systems consisting of tens of thousands of processors, with near perfect O(N) scaling with system size and wall times as low as a few seconds per self-consistent field iteration. Finally, we verify the accuracy of SQDFT in large-scale quantum molecular dynamics simulations of aluminum at high temperature.

M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations

“…KSSOLV [10] is one such Matlab code for the planewave method, traditionally the discretization of choice in Kohn-Sham DFT [11,12,13,14,15,16]. However, to the best of our knowledge, no such counterpart exists for real-space methods, which have gained significant attention recently [17,18,19,20,21,22,23,24], in part due to their high scalability for large-scale parallel computing [25,23,24], flexibility in the choice of boundary conditions [26,27,28], and amenability to the development of linear scaling methods [29,8]. Motivated by this, in this work we develop M-SPARC: Matlab-Simulation Package for Ab-initio Realspace Calculations.…”

Spectral quadrature for the first principles study of crystal defects: Application to magnesium

“…It also showed that purification methods using Chebyshev polynomials are equivalent to the use of Clenshaw-Curtis quadratures. This understanding led to a series of efficient algorithms using spectral quadratures [45,58] and applications to first-principles molecular dynamics [58,67,50].…”

“…The first goal of this work is to retain the efficiency of LSSGQ, but also extend it to nonlocal pseudopotentials. This requires a methodological advance that combines ideas of Ponga et al [42] and Suryanarayana et al [58]. As noted above, LSSGQ efficiently computes the energy and electron density using the Lanczos iteration since these only depend on the diagonal component of the density matrix.…”

“…The parallelization approach of Ponga et al [42] required example-specific assignment; instead we use domain decomposition enabling easier implementation on HPC platforms. The implementation of Suryanarayana et al [58] was limited to periodic boundary conditions and cuboidal domains. These are not appropriate for the study of defects.…”

On preconditioning the self-consistent field iteration in real-space Density Functional Theory

“…Since these preconditioners have a diagonal representation in Fourier space, they are easy and efficient to apply within the planewave method, but are unsuitable for real-space methods, where they take a global form. Given that realspace codes are now able to outperform their planewave counterparts by being able to leverage large-scale computational resources [26,27], while being amenable to the development of linearscaling methods [28,29] and offering the flexibility in choice of boundary conditions [30,31,32], efficient real-space analogues for such preconditioners are highly desired.…”