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تسجيل مستخدم جديدDistributed-memory $mathcal{H}$-matrix Algebra I: Data Distribution and Matrix-vector Multiplication
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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 $Omega(P^2)$ 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 $OBig(frac{N log N}{P} + alpha log P + beta log^2 P Big)$ for $mathcal{H}$-matrices under weak admissibility condition, where $N$ is the matrix size, $alpha$ denotes the latency, and $beta$ 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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