Sparse approximate matrix multiplication in a fully recursive distributed task-based parallel framework

In this paper we consider parallel implementations of approximate multiplication of large matrices with exponential decay of elements. Such matrices arise in computations related to electronic structure calculations and some other fields of science. Commonly, sparsity is introduced by truncation of input matrices. In turn, the sparse approximate multiplication algorithm [M. Challacombe and N. Bock, arXiv preprint 1011.3534, 2010] performs truncation of sub-matrix products. We consider these two methods and their combination, i.e. truncation of both input matrices and sub-matrix products. Implementations done using the Chunks and Tasks programming model and library [E. H. Rubensson and E. Rudberg, Parallel Comput., 40:328343, 2014] are presented and discussed. The absolute error asymptotic behavior is derived. A comparison between the three methods in terms of performance is done on a model problem. The algorithms are also applied to matrices coming from large chemical systems with $\approx 10^6$ atoms.