Efficient Shift-and-Invert Preconditioning for Multi-GPU Accelerated Density Functional Calculations.

To accelerate the iterative diagonalization of electronic structure calculations, we propose a new inexact shift-and-invert (ISI) preconditioning method. The key idea is to improve shift values in the ISI preconditioning to be closer to the exact eigenvalues, leading to a significant boost in the convergence speed of the iterative diagonalization. Furthermore, we adopted a preconditioned conjugate gradient solver to rapidly evaluate an inversion process. Finally, we accelerated overall processes, including the proposed modification, with state-of-the-art graphical processing units (GPUs) and assessed its parallel efficiency with real-space density functional calculations of 1D, 2D, and 3D periodic systems. Our method attains both fast diagonalization convergence and high multi-GPU parallel efficiency. This is evident from the fact that single-point density functional calculations for hundreds of atom systems can be done in approximately 10 s using 8 GPUs. The proposed method can be generally applied to any electronic structure calculation methods involving large-scale diagonalizations.