Parallel One–Sided Jacobi SVD Algorithm with Variable Blocking Factor

Bečka, Martin; Okša, Gabriel

doi:10.1007/978-3-642-55224-3_6

Cited by 6 publications

(20 citation statements)

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“…The algorithm consists of preprocessing (lines 1-3), iteration (lines [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], and postprocessing (lines [20][21][22][23][24][25]. Here, > 0 is a predetermined convergence criterion and a r i denotes the ith column vector of A .r/ .…”

Section: The Algorithmmentioning

confidence: 99%

“…As for the ordering, we focus on the so-called parallel dynamic ordering strategy [7]. As for the matrix distribution over computing nodes, we assume variable blocking [8], in which the blocks A .r/ i (i D 1; 2; : : : ;`) are divided into column subblocks and each subblock is allocated to one computing node. It has been shown that this strategy is effective in reducing the number of iterations of the OSBJ method for a wide class of matrices, including various random matrices [13,14] and matrices from real applications [7].…”

Section: Introductionmentioning

confidence: 99%

“…In this strategy, at each step r, one chooses`=2 independent pairs .i r;1 ; j r;1 /, : : :, .i r;`=2 ; j r;`=2 / by taking into account the mutual perpendicularity of all pairs of block columns, and then orthogonalize the columns of the`=2 submatrices OEA .r/ i r;1 ; A .r/ j r;1 ; : : : ; OEA .r/ i r;`=2 ; A .r/ j r;`=2 in parallel. The approach is shown [8] to achieve better parallel efficiency than the conventional fixed-size blocking, where the pair OEA .r/ i ; A .r/ j is allocated to one node. As for the matrix distribution over computing nodes, we assume variable blocking [8], in which the blocks A .r/ i (i D 1; 2; : : : ;`) are divided into column subblocks and each subblock is allocated to one computing node.…”

Section: Introductionmentioning

confidence: 99%

“…As the level-2 BLAS is a memory-intensive operation and cannot use cache memory efficiently, this tends to lower the performance, especially when the matrix size is large.An alternative to the bi-diagonalization-based method is the one-sided block Jacobi method (OSBJ) [5][6][7][8][9][10][11][12], which recently attracted attention. In this paper, we focus on the most recent variant of the OSBJ method, the one with parallel dynamic ordering and variable blocking, and present both theoretical and experimental analyses of the algorithm.…”

mentioning

confidence: 99%

“…An alternative to the bi-diagonalization-based method is the one-sided block Jacobi method (OSBJ) [5][6][7][8][9][10][11][12], which recently attracted attention. In this method, one first partitions the input matrix logically into block columns as ‡ In the main loop, about`=.`C 1/ of the total computational work is performed in the form of level-3 BLAS.…”

mentioning

confidence: 99%

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Performance analysis and optimization of the parallel one‐sided block Jacobi SVD algorithm with dynamic ordering and variable blocking

Kudo

Yamamoto

Bečka

et al. 2016

Concurrency and Computation

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The one-sided block Jacobi (OSBJ) method is known to be an efficient method for computing the singular value decomposition on a parallel computer. In this paper, we focus on the most recent variant of the OSBJ method, the one with parallel dynamic ordering and variable blocking, and present both theoretical and experimental analyses of the algorithm. In the first part of the paper, we provide a detailed theoretical analysis of its convergence properties. In the second part, based on preliminary performance measurement on the Fujitsu FX10 and SGI Altix ICE parallel computers, we identify two performance bottlenecks of the algorithm and propose new implementations to resolve the problem. Experimental results show that they are effective and can achieve up to 1.8 and 1.4 times speedup of the total execution time on the FX10 and the Altix ICE, respectively. Comparison with the ScaLAPACK SVD routine PDGESVD shows that our OSBJ solver is efficient when solving small to medium sized problems (n < 10000) using modest number (< 100) of computing nodes. this approach. However, from the viewpoint of high performance computing, this approach has two drawbacks. First, the bi-diagonalization step has fine-grained parallelism and requires O.n/ interprocessor communications. This often causes performance bottleneck. Second, half of the computational work in the bi-diagonalization step is done in the form of level-2 BLAS (matrix-vector multiplication). As the level-2 BLAS is a memory-intensive operation and cannot use cache memory efficiently, this tends to lower the performance, especially when the matrix size is large.An alternative to the bi-diagonalization-based method is the one-sided block Jacobi method (OSBJ) [5][6][7][8][9][10][11][12], which recently attracted attention. In this method, one first partitions the input matrix logically into block columns as ‡ In the main loop, about`=.`C 1/ of the total computational work is performed in the form of level-3 BLAS. See

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