This article introduces HODLR3D, a class of hierarchical matrices arising out of $N$-body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the $N$-body problems in three dimensions are numerically low-rank. For the Laplace kernel in $3$D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in $3$D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for $N$-body problems with large system sizes.

翻译：本文介绍HODLR3D，这是一类源于三维N体问题的层级矩阵。HODLR3D基于如下事实：三维N体问题中产生的某些非对角矩阵子块在数值上具有低秩性。针对广泛存在的三维拉普拉斯核，我们证明了在有限精度下所有非对角矩阵子块均为秩亏缺，并获得了这些子块秩随其尺寸增长的规律。对于其他三维核函数，我们通过数值实验展示了不同非对角子块具有类似的秩增长特性。我们利用这种层级低秩结构构建了HODLR3D表示，并以此加速矩阵-向量乘积运算。HODLR3D矩阵-向量乘积的存储与计算复杂度与系统规模呈近线性关系。通过多项数值实验，我们验证了HODLR3D表示的计算性能。此外，我们还探讨了HODLR3D表示在分布式内存系统上的表现。本文所述HODLR3D基于弱可容许性条件。在三维具有不同弱可容许性条件的层级矩阵中，仅HODLR3D的可容许非对角块秩不随系统规模的任意次幂增长。因此，对于大规模N体问题，HODLR3D矩阵-向量乘积的存储与计算复杂度始终保持可处理性。