HODLR3D: hierarchical matrices for N-body problems in three dimensions

Abstract

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 3D, 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 3D, 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.

Appendices

Appendix A: Numerical experiment on HODLR3D matrix-vector product

Fig. 14

Various benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and $\mathcal {H}_{\sqrt{3}}$ matrix-vector products for the kernel $\frac{1}{r}$ with the relative forward errors of the three algorithms to be of the same order

Fig. 15

Various benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and $\mathcal {H}_{\sqrt{3}}$ matrix-vector products for the kernel $\frac{1}{r^4}$ with the relative forward errors of the three algorithms to be of the same order

In this section, we repeat the numerical experiment in Section 4.1 for the different hierarchical structures considered, viz., HODLR, HODLR3D, and $\mathcal {H}_{\sqrt{3}}$ matrix such that in matrix-vector product the forward relative error is of the same order. This numerical experiment intends to understand the performance and scalability of different hierarchical structures for the same matrix-vector product forward relative error. We make sure that the relative forward error of the three algorithms that we compare, HODLR3D, HODLR, and $\mathcal {H}_{\sqrt{3}}$ matrix, are nearly equal so that the rest of the benchmarks can be compared and an inference can be made. To achieve this, we use different values of $\epsilon$ in the ACA routine of the three hierarchical structures, in the range of $10^{-6}-10^{-10}$. We perform this incrementally and record various benchmarks of the hierarchical structures, such that they have a forward relative error of the same order. The kernels that we use to perform the numerical experiment are as follows:

• Green’s function for Laplace equation in 3D which is $\dfrac{1}{r}$

• $\dfrac{1}{r^4}$

• Real part of Green’s function for Helmholtz equation in 3D, which is $\dfrac{\cos \left( r\right) }{r}$

From Figs. 14, 15, and 16, we observe that by maintaining the relative forward error to be nearly equal, the computational complexity for the matrix-vector product using HODLR3D and $\mathcal {H}_{\sqrt{3}}$-matrix representation still roughly scales $\mathcal {O} \left( N\log \left( N\right) \right)$, which is not the case with HODLR in 3D.

Fig. 16

Various benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and $\mathcal {H}_{\sqrt{3}}$ matrix-vector products for the kernel $\frac{\cos \left( r\right) }{r}$ with the relative forward errors of the three algorithms to be of the same order

Appendix B: HODLR3D initialization in distributed memory systems

As discussed in Section 6, we consider the nodes in a particular level of the hierarchical tree as data-independent computational units. For level l, where the number of nodes in a level $\left( 8^l\right)$ is greater than $n_p$ MPI processes, each MPI process has $\left\lceil \dfrac{8^l}{n_p} \right\rceil$ computational units. For level l, where the number of nodes in that level is lesser than $n_p$ MPI processes, each node in level l is shared by $\left\lceil \dfrac{n_p}{8^l} \right\rceil$ MPI processes. The low-rank compression involved with the shared node is performed separately by each MPI process that shares that node. This is performed to eliminate the communication involved and reduce idle time. Table 9 in the Appendix shows the time taken by parallel HODLR3D to initialize the data structure. Additionally, the scalability is ideal when $8^l>n_p$. However, the scalability of HODLR3D initialization is limited by the construction of the low-rank approximation corresponding to the node in the hierarchical tree at level l, where $8^l<n_p$.

Table 9 Parallel HODLR3D initialization