Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures

Abstract

Quantum mechanical calculations for material modelling using Kohn-Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for $N$ smallest eigenvector-eigenvalue pairs with $N$ proportional to the number of electrons in the material system. These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn-Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh-Ritz projection step of the discretized Kohn-Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node CPU and GPU architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behaviour and energy efficiency.

arXiv comment10 pages, 7 figures

References in corpus (14)

• The Kernel Polynomial Method
• DFT-FE -- A massively parallel adaptive finite-element code for large-scale density functional theory calculations
• Expansion algorithm for the density matrix
• Higher-order adaptive finite-element methods for Kohn-Sham density functional theory
• SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters
• DFT-FE 1.0: A massively parallel hybrid CPU-GPU density functional theory code using finite-element discretization
• GPU-Acceleration of the ELPA2 Distributed Eigensolver for Dense Symmetric and Hermitian Eigenproblems
• A subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization
• Implicit Purification for Temperature-Dependent Density Matrices
• Large scale quantum chemistry with Tensor Processing Units
• Non-monotonic recursive polynomial expansions for linear scaling calculation of the density matrix
• Generalized canonical purification for density matrix minimization
• Mixed Precision Fermi-Operator Expansion on Tensor Cores From a Machine Learning Perspective
• Red-blue pebbling revisited: near optimal parallel matrix-matrix multiplication