Dynamic Precision Approach for Accelerating Large-Scale Eigenvalue Solvers in Electronic Structure Calculations on Graphics Processing Units

Abstract

Single precision (SP) arithmetic can be greatly accelerated as compared to double precision (DP) arithmetic on graphics processing units (GPUs). However, the use of SP in the whole process of electronic structure calculations is inappropriate for the required accuracy. We propose a 3-fold dynamic precision approach for accelerated calculations but still with the accuracy of DP. Here, SP, DP, and mixed precision are dynamically switched during an iterative diagonalization process. We applied this approach to the locally optimal block preconditioned conjugate gradient method to accelerate a large-scale eigenvalue solver for the Kohn–Sham equation. We determined a proper threshold for switching each precision scheme by examining the convergence pattern on the eigenvalue solver only with the kinetic energy operator of the Kohn–Sham Hamiltonian. As a result, we achieved up to 8.53× and 6.60× speedups for band structure and self-consistent field calculations, respectively, for test systems under various boundary conditions on NVIDIA GPUs.

Gaussian-Approximated Poisson Preconditioner for Iterative Diagonalization in Real-Space Density Functional Theory

“…The iterative diagonalization of the Hamiltonian matrix is a computational bottleneck in real-space DFT calculations. To facilitate efficient diagonalization of the Hamiltonian, various iterative algorithms have been applied, − among which preconditioned iterative methods have been widely adopted in many real-space simulation codes. ,,,,, Despite recent advancements in iterative eigensolvers for real-space DFT, the overall efficiency is still hindered by the absence of efficient preconditioners.…”

“…The experiments in this study used a convergence threshold of 1e-3, so the residual sizes in the figures were plotted up to 1e-3, which corresponds to an eigenvalue error of about 1 mHartree. 38 Concerns may arise regarding the possible variation in the convergence behavior of the preconditioner at lower residual size. To address this, we conducted additional experiments on the same systems in Figure 2, using a lower convergence threshold of 1e-7 (see Figure S3).…”