A fast, dense Chebyshev solver for electronic structure on GPUs

TL;DR

This paper introduces a GPU-optimized Chebyshev expansion algorithm for electronic structure calculations, achieving significant speedups over traditional diagonalization for modest matrix sizes in quantum chemistry.

Contribution

It derives and implements a Chebyshev expansion method tailored for GPUs, improving performance and parallelism over existing algorithms for dense matrices of modest size.

Findings

Achieves large speedups compared to diagonalization on GPUs.

Improves upon existing Chebyshev methods by leveraging task parallelism.

Successfully applied to a high-density system around the Fermi level.

Abstract

Matrix diagonalization is almost always involved in computing the density matrix needed in quantum chemistry calculations. In the case of modest matrix sizes ($\lesssim$ 5000), performance of traditional dense diagonalization algorithms on modern GPUs is underwhelming compared to the peak performance of these devices. This motivates the exploration of alternative algorithms better suited to these types of architectures. We newly derive, and present in detail, an existing Chebyshev expansion algorithm [W. Liang et al, J. Chem. Phys. 2003] whose number of required matrix multiplications scales with the square root of the number of terms in the expansion. Focusing on dense matrices of modest size, our implementation on GPUs results in large speed ups when compared to diagonalization. Additionally, we improve upon this existing method by capitalizing on the inherent task parallelism and concurrency in the algorithm. This improvement is implemented on GPUs by using CUDA and HIP streams via the MAGMA library and leads to a significant speed up over the serial-only approach for smaller ($\lesssim$ 1000) matrix sizes. Lastly, we apply our technique to a model system with a high density of states around the Fermi level which typically presents significant challenges.