Sparse approximate matrix-matrix multiplication for density matrix purification with error control

Anton G Artemov,Emanuel H Rubensson

doi:10.1016/j.jcp.2021.110354

Abstract

We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening of submatrix products in the block-sparse matrix-matrix multiplies to reduce the computational cost. An error bound and a parameter sweep are combined to select a threshold value for the screening, such that the error can be controlled. We evaluate the performance of the method in comparison to purification without acceleration and without submatrix product screening.

Highlights

Sparse matrix-matrix multiplication is a key operation in linear scaling electronic structure calculations based on, for example, Hartree–Fock or Kohn–Sham density functional theory
We propose a new purification algorithm based on the second order spectral projection method (SP2) [17] using the scale-and-fold acceleration technique [14,19,20] and an error control that accounts for errors due to submatrix product screening
We propose here an accelerated density matrix purification scheme with error control and prescreening of small submatrix products in the sparse matrix-matrix multiplications

Summary

Introduction

Sparse matrix-matrix multiplication is a key operation in linear scaling electronic structure calculations based on, for example, Hartree–Fock or Kohn–Sham density functional theory This operation has received a lot of attention from method and software developers in this field [6]. Many of the just introduced nonzeros have small magnitude and will anyway be removed in the subsequent truncation This means that computational resources are used to compute and temporarily store those matrix entries for no purpose. In the cutoff radius approach all matrix entries that correspond to distances between nuclei or basis function centers larger than some cutoff radius are excluded from the representation [5,18] Since, in this case, the nonzero pattern is known in advance, the product may be computed directly in truncated form. We make use of the SpAMM algorithm but add a preceding step to carefully select the SpAMM tolerance to achieve error control in the whole purification scheme

Error control in SpAMM

19: Errors

Accelerated density matrix purification with error control

Method n

Experimental evaluation

Concluding remarks