CSIAM Trans. Appl. Math., 2 (2021), pp. 431-459.

Published online: 2021-08

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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 $\mathscr{O}(\frac{Nlog N}{P} +αlog P+βlog^2P)$ 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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0206}, url = {http://global-sci.org/intro/article_detail/csiam-am/19445.html} }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 $\mathscr{O}(\frac{Nlog N}{P} +αlog P+βlog^2P)$ 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.

*CSIAM Transactions on Applied Mathematics*.

*2*(3). 431-459. doi:10.4208/csiam-am.2020-0206