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

TL;DR

This paper investigates recursive Fermi-operator expansion strategies as an alternative to traditional diagonalization methods for large-scale DFT eigenvalue problems, aiming to improve computational efficiency on HPC architectures.

Contribution

It introduces and compares polynomial expansion approaches to reduce the computational cost of solving large-scale Kohn-Sham DFT eigenvalue problems, especially on parallel architectures.

Findings

Recursive polynomial expansion can outperform diagonalization in large systems.

The methods show favorable scaling and energy efficiency on HPC architectures.

Performance varies depending on the hardware and system size.

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.