Volume 2, Issue 3
Distributed-Memory $\mathcal{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication

Yingzhou Li, Jack Poulson & Lexing Ying

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

Published online: 2021-08

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  • Abstract

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.

  • Keywords

Parallel fast algorithm, $\mathcal{H}$-matrix, distributed-memory, parallel computing.

  • AMS Subject Headings

65F99, 65Y05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-431, author = {Yingzhou and Li and and 18064 and and Yingzhou Li and Jack and Poulson and and 18065 and and Jack Poulson and Lexing and Ying and and 18066 and and Lexing Ying}, title = {Distributed-Memory $\mathcal{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {3}, pages = {431--459}, abstract = {

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} }
TY - JOUR T1 - Distributed-Memory $\mathcal{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication AU - Li , Yingzhou AU - Poulson , Jack AU - Ying , Lexing JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 431 EP - 459 PY - 2021 DA - 2021/08 SN - 2 DO - http://doi.org/10.4208/csiam-am.2020-0206 UR - https://global-sci.org/intro/article_detail/csiam-am/19445.html KW - Parallel fast algorithm, $\mathcal{H}$-matrix, distributed-memory, parallel computing. AB -

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.

Yingzhou Li, Jack Poulson & LexingYing. (2021). Distributed-Memory $\mathcal{H}$-Matrix Algebra I: Data Distribution and Matrix-Vector Multiplication. CSIAM Transactions on Applied Mathematics. 2 (3). 431-459. doi:10.4208/csiam-am.2020-0206
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