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Title: Distributed-memory lattice H-matrix factorization

Abstract

We parallelize the LU factorization of a hierarchical low-rank matrix (H-matrix) on a distributed-memory computer. This is much more difficult than the H-matrix-vector multiplication due to the dataflow of the factorization, and it is much harder than the parallelization of a dense matrix factorization due to the irregular hierarchical block structure of the matrix. Block low-rank (BLR) format gets rid of the hierarchy and simplifies the parallelization, often increasing concurrency. However, this comes at a price of losing the near-linear complexity of the H-matrix factorization. In this work, we propose to factorize the matrix using a “lattice H-matrix” format that generalizes the BLR format by storing each of the blocks (both diagonals and off-diagonals) in the H-matrix format. These blocks stored in the H-matrix format are referred to as lattices. Thus, this lattice format aims to combine the parallel scalability of BLR factorization with the near-linear complexity of H-matrix factorization. We first compare factorization performances using the H-matrix, BLR, and lattice H-matrix formats under various conditions on a shared-memory computer. Our performance results show that the lattice format has storage and computational complexities similar to those of the H-matrix format, and hence a much lower cost of factorization than BLR.more » In conclusion, we then compare the BLR and lattice H-matrix factorization on distributed-memory computers. Our performance results demonstrate that compared with BLR, the lattice format with the lower cost of factorization may lead to faster factorization on the distributed-memory computer.« less

Authors:
ORCiD logo [1];  [2]; ORCiD logo [3];  [4]
+ Show Author Affiliations
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. The Univ. of Tokyo, Tokyo (Japan)
  3. Tokyo Inst. of Technology, Tokyo (Japan)
  4. The Univ. of Tennessee, Knoxville, TN (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1559494
Report Number(s):
SAND-2019-8102J
Journal ID: ISSN 1094-3420; 677691
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
International Journal of High Performance Computing Applications
Additional Journal Information:
Journal Volume: 33; Journal Issue: 5; Journal ID: ISSN 1094-3420
Publisher:
SAGE
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; boundary element method; LU factorization; distributed memory; hierarchical matrix; task programming

Citation Formats

Yamazaki, Ichitaro, Ida, Akihiro, Yokota, Rio, and Dongarra, Jack. Distributed-memory lattice H-matrix factorization. United States: N. p., 2019. Web. doi:10.1177/1094342019861139.
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Yamazaki, Ichitaro, Ida, Akihiro, Yokota, Rio, & Dongarra, Jack. Distributed-memory lattice H-matrix factorization. United States. https://doi.org/10.1177/1094342019861139
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Yamazaki, Ichitaro, Ida, Akihiro, Yokota, Rio, and Dongarra, Jack. Thu . "Distributed-memory lattice H-matrix factorization". United States. https://doi.org/10.1177/1094342019861139. https://www.osti.gov/servlets/purl/1559494.
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@article{osti_1559494,
title = {Distributed-memory lattice H-matrix factorization},
author = {Yamazaki, Ichitaro and Ida, Akihiro and Yokota, Rio and Dongarra, Jack},
abstractNote = {We parallelize the LU factorization of a hierarchical low-rank matrix (H-matrix) on a distributed-memory computer. This is much more difficult than the H-matrix-vector multiplication due to the dataflow of the factorization, and it is much harder than the parallelization of a dense matrix factorization due to the irregular hierarchical block structure of the matrix. Block low-rank (BLR) format gets rid of the hierarchy and simplifies the parallelization, often increasing concurrency. However, this comes at a price of losing the near-linear complexity of the H-matrix factorization. In this work, we propose to factorize the matrix using a “lattice H-matrix” format that generalizes the BLR format by storing each of the blocks (both diagonals and off-diagonals) in the H-matrix format. These blocks stored in the H-matrix format are referred to as lattices. Thus, this lattice format aims to combine the parallel scalability of BLR factorization with the near-linear complexity of H-matrix factorization. We first compare factorization performances using the H-matrix, BLR, and lattice H-matrix formats under various conditions on a shared-memory computer. Our performance results show that the lattice format has storage and computational complexities similar to those of the H-matrix format, and hence a much lower cost of factorization than BLR. In conclusion, we then compare the BLR and lattice H-matrix factorization on distributed-memory computers. Our performance results demonstrate that compared with BLR, the lattice format with the lower cost of factorization may lead to faster factorization on the distributed-memory computer.},
doi = {10.1177/1094342019861139},
journal = {International Journal of High Performance Computing Applications},
number = 5,
volume = 33,
place = {United States},
year = {Thu Aug 01 00:00:00 EDT 2019},
month = {Thu Aug 01 00:00:00 EDT 2019}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1177/1094342019861139
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Citation Metrics:
Cited by: 11 works
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Figures / Tables:

Fig. 1 Fig. 1: Conceptual diagram of the construction of representative low-rank structured matrices using different partition structure M: (a) $\mathscr{M_H}$ for a general $\mathscr{H}$-matrix, (b) $\mathscr{M}$W for a Hierarchical Semi-Separable (HSS), (c) $\mathcal{M}$L for a Block Low-Rank (BLR), and (d) $\mathscr{M}$L$\mathscr{H}$for a lattice $\mathscr{H}$-matrix. Blank boxes show non-admissible blocks. Blocks inmore » light red indicate submatrices judged as lowrank, and blocks painted in deep red are remaining non-admissible blocks calculated as dense submatrices.« less
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