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Title: Distributed-memory lattice H-matrix factorization

Journal Article · · International Journal of High Performance Computing Applications
ORCiD logo [1];  [2]; ORCiD logo [3];  [4]
  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)
+ Show Author Affiliations

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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC04-94AL85000
OSTI ID:
1559494
Report Number(s):
SAND-2019-8102J; 677691
Journal Information:
International Journal of High Performance Computing Applications, Vol. 33, Issue 5; ISSN 1094-3420
Publisher:
SAGECopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 11 works
Citation information provided by
Web of Science

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Figures / Tables (24)

Fig. 1(p. 4)figure Fig. 1
Table 1(p. 6)table Table 1
Fig. 2(p. 7)figure Fig. 2
Fig. 3(p. 7)figure Fig. 3
Fig. 4(p. 8)figure Fig. 4
Fig. 5(p. 8)figure Fig. 5
Fig. 6(p. 9)figure Fig. 6
Fig. 7(p. 9)figure Fig. 7
r is assumed to be constant, and it does not appear in the complexity estimates. The cost of re-compressing the low-rank blocks is included in gemm update (e.g., $$\mathscr{O}$$($$\mathscr{ℓr}$$) FLOPs for recompression, compared with $$\mathscr{O}$$($$\mathscr{ℓr}^2$$) FLOPs for update)." data-ostiid="1559494">
Table 2(p. 11)table Table 2
Fig. 8(p. 12)figure Fig. 8
Fig. 9(p. 12)figure Fig. 9
Fig. 10(p. 13)figure Fig. 10
Fig. 11(p. 13)figure Fig. 11
Table 3(p. 13)table Table 3
Fig.12(p. 14)figure Fig.12
9∥b -Ax∥/(n∥b∥) and 102∙ ∥x-$$\bar{x}$$∥/∥x∥, respectively, while GB(A) and GB(LU) are the respective storage costs for the original $$\mathscr{H}$$-matrix and its LU factors in Gigabytes. BLU failed for 212t and 338t to excessive memory." data-ostiid="1559494">
Table 4(p. 14)table Table 4
Table 5(p. 14)table Table 5
Fig. 13(p. 15)figure Fig. 13
Fig. 14(p. 15)figure Fig. 14
Fig. 15(p. 15)figure Fig. 15
Fig. 16(p. 16)figure Fig. 16
Table 6(p. 16)table Table 6
Table 7(p. 16)table Table 7
Table 8(p. 17)table Table 8

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Related Subjects

97 MATHEMATICS AND COMPUTING
boundary element method
LU factorization
distributed memory
hierarchical matrix
task programming