Parallel Sparse Matrix Multiplication for Linear Scaling Electronic Structure Calculations

Abstract

Linear-scaling electronic-structure techniques, also called O(N) techniques, rely heavily on the multiplication of sparse matrices, where the sparsity arises from spatial cut-offs. In order to treat very large systems, the calculations must be run on parallel computers. We analyse the problem of parallelising the multiplication of sparse matrices with the sparsity pattern required by linear-scaling techniques. We show that the management of inter-node communications and the effective use of on-node cache are helped by organising the atoms into compact groups. We also discuss how to identify a key part of the code called the `multiplication kernel', which is repeatedly invoked to build up the matrix product, and explicit code is presented for this kernel. Numerical tests of the resulting multiplication code are reported for cases of practical interest, and it is shown that their scaling properties for systems containing up to 20,000 atoms on machines having up to 512 processors are excellent. The tests also show that the cpu efficiency of our code is satisfactory.

20 pages, 6 figures, submitted to Computer Physics Communications

References in corpus (1)

• An Efficient and Robust Technique for Achieving Self Consistency in Electronic Structure Calculations

Cited by in corpus (12)

• O(N) methods in electronic structure calculations
• Recent progress with large-scale ab initio calculations: the CONQUEST code
• Calculations on millions of atoms with DFT: Linear scaling shows its potential
• Expansion algorithm for the density matrix
• Next generation extended Lagrangian first principles molecular dynamics
• Optimized multi-site local orbitals in the large-scale DFT program CONQUEST
• Efficient Calculations with Multisite Local Orbitals in the large-scale DFT Code CONQUEST
• Highly accurate local basis sets for large-scale DFT calculations in CONQUEST
• Locality-aware parallel block-sparse matrix-matrix multiplication using the Chunks and Tasks programming model
• Canonical-ensemble extended Lagrangian Born-Oppenheimer molecular dynamics for the linear scaling density functional theory
• The pseudoatomic orbital basis: electronic accuracy and soft-mode distortions in ABO${}_3$ perovskites
• Sparse approximate matrix-matrix multiplication for density matrix purification with error control