Space-Filling Curves for Real-Space Electronic Structure Calculations

Hamiltonian matrices for Kohn-Sham calculations implemented in real space are often large (millions by millions) but very sparse. This poses challenges and opportunities for iterative eigensolvers, which often require a large number of matrix-vector multiplications. As a consequence, an efficient parallel sparse matrix-vector multiplication algorithm is desired. Here, we investigate the benefits of using Hilbert space-filling curves (SFCs) in domain partitioning. We show that the use of Hilbert SFCs in grid-point partitioning brings better locality of the grid points, improves balance of communication, and reduces communication overhead. We also demonstrate an extension of Hilbert SFCs coupled with blockwise operations. The use of blockwise operations helps exploit the vector-processing units in contemporary computational platforms. We illustrate speedup and scalability improvements for an iterative eigensolver based on the Chebyshev-filtered subspace iteration method. Using blockwise Hilbert SFCs, we solve the Kohn-Sham problem for silicon nanocrystals up to 10 nm in diameter, which contain over 26,000 atoms. We illustrate how the density of states of silicon nanocrystals evolves to the bulk limit, where Van Hove singularities are clearly apparent.