分布存储$\mathcal{H}$ -矩阵代数I：数据分布与矩阵向量乘法
Distributed-memory $\mathcal{H}$-matrix Algebra I: Data Distribution and Matrix-vector Multiplication
摘要：
介绍了$\mathcal{H}$ -矩阵的数据分布方案和$\mathcal{H}$ -矩阵-向量乘法的分布存储算法。我们的数据分布方案避免了昂贵的$Ω(P^2)$调度过程中使用的在以前的工作，其中$P$是进程的数量，而数据平衡得到很好的保护。该算法根据数据分布的特点，将计算均匀地分配到各个$P$进程中，并采用一种新的树型通信算法来降低延迟代价。对于弱容许条件下的$\mathcal{H}$ -矩阵，我们的算法的总复杂度为$O\Big(\frac{N \log N}{P} + α\log P + β\log^2 P \Big)$，其中$N$是矩阵大小，$α$表示延迟，$β$表示逆带宽.数值上，我们的算法适用于解决各种数量的进程之间的各种大小的二维和三维问题。在数千个进程上，仍然可以观察到良好的并行效率。
英文摘要：
We introduce a data distribution scheme for $\mathcal{H}$-matrices and a distributed-memory algorithm for $\mathcal{H}$-matrix-vector multiplication. Our data distribution scheme avoids an expensive $Ω(P^2)$ scheduling procedure used in previous work, where $P$ is the number of processes, while data balancing is well-preserved. Based on the data distribution, our distributed-memory algorithm evenly distributes all computations among $P$ processes and adopts a novel tree-communication algorithm to reduce the latency cost. The overall complexity of our algorithm is $O\Big(\frac{N \log N}{P} + α\log P + β\log^2 P \Big)$ for $\mathcal{H}$-matrices under weak admissibility condition, where $N$ is the matrix size, $α$ denotes the latency, and $β$ denotes the inverse bandwidth. Numerically, our algorithm is applied to address both two- and three-dimensional problems of various sizes among various numbers of processes. On thousands of processes, good parallel efficiency is still observed.
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