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In this paper we consider parallel implementations of approximatemultiplication of large matrices with exponential decay of elements. Suchmatrices arise in computations related to electronic structure calculations andsome other fields of computational science. Commonly, sparsity is introduced bydropping out small entries (truncation) of input matrices. Another approach,the sparse approximate multiplication algorithm [M. Challacombe and N. Bock,arXiv preprint 1011.3534, 2010] performs truncation of sub-matrix products. Weconsider these two methods and their combination, i.e. truncation of both inputmatrices and sub-matrix products. Implementations done using the Chunks andTasks programming model and library [E. H. Rubensson and E. Rudberg, ParallelComput., 40:328-343, 2014] are presented and discussed. We show that theabsolute error in the Frobenius norm behaves as $O\left(n^{1/2} \right), n\longrightarrow \infty $ and $O\left(\tau^{p/2} \right), \tau \longrightarrow0,\,\, \forall p < 2$ for all three methods, where $n$ is the matrix size and$\tau$ is the truncation threshold. We compare the methods on a model problemand show that the combined method outperforms the original two. The methods arealso applied to matrices coming from large chemical systems with $\sim 10^6$atoms. We show that the combination of the two methods achieves better weakscaling by reducing the amount of communication by a factor of $\approx 2$.

Approximate multiplication of nearly sparse matrices with decay in a fully recursive distributed task-based parallel framework