在本文中，我们考虑并行实现的近似乘法的大型矩阵与指数衰减的元素。这种矩阵出现在与电子结构计算和计算科学的一些其他领域相关的计算中。通常，稀疏性通过丢弃输入矩阵的小条目（截断）来引入。另一种方法是稀疏近似乘法算法[M. Challacombe和N. Bock，arXiv preprint 1011.3534，2010]执行子矩阵乘积的截断。我们考虑这两种方法及其组合，即截断输入矩阵和子矩阵乘积。使用Chunks and Tasks编程模型和库完成的实现[E. H. Rubensson和E. Rudberg，并行计算，40：328-343，2014]被呈现和讨论。我们表明，在Frobenius范数的绝对误差表现为$O\left(n^{1/2} \right), n \longrightarrow \infty $和$O\left(τ^{p/2} \right), τ\longrightarrow 0,\,\, \forall p < 2$的所有三种方法，其中$n$是矩阵的大小和$τ$是截断阈值。我们比较了一个模型问题的方法，并表明，合并后的方法优于原来的两个。该方法也适用于矩阵来自大型化学系统与$\sim 10^6$原子。我们表明，这两种方法的结合，实现了更好的弱缩放，通过减少通信量的一个因素的$\approx 2$。
英文摘要：
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 computational science. Commonly, sparsity is introduced by dropping 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. 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., 40:328-343, 2014] are presented and discussed. We show that the absolute error in the Frobenius norm behaves as $O\left(n^{1/2} \right), n \longrightarrow \infty $ and $O\left(τ^{p/2} \right), τ\longrightarrow 0,\,\, \forall p < 2$ for all three methods, where $n$ is the matrix size and $τ$ is the truncation threshold. We compare the methods on a model problem and show that the combined method outperforms the original two. The methods are also applied to matrices coming from large chemical systems with $\sim 10^6$ atoms. We show that the combination of the two methods achieves better weak scaling by reducing the amount of communication by a factor of $\approx 2$.
Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions
Center for Computational Mathematics, Flatiron Institute, Simons Foundation（计算数学中心，Flatiron研究所，西蒙斯基金会）；Institute for Theoretical Sciences, Westlake University（理论科学研究所，西湖大学）
