Communication Avoiding Block Low-Rank Parallel Multifrontal Triangular Solve with Many Right-Hand Sides

Block low-rank (BLR) compression can significantly reduce the memory and time costs of parallel sparse direct solvers. In this paper, we investigate the performance of the BLR triangular solve phase, which we observe to be underwhelming when dealing with many right-hand sides (RHS). We explain that...

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Vydáno v:SIAM journal on matrix analysis and applications Ročník 45; číslo 1; s. 148 - 166
Hlavní autoři: Amestoy, Patrick, Boiteau, Olivier, Buttari, Alfredo, Gerest, Matthieu, Jézéquel, Fabienne, L’Excellent, Jean-Yves, Mary, Theo
Médium: Journal Article
Jazyk:angličtina
Vydáno: Society for Industrial and Applied Mathematics 01.01.2024
Témata:
Computer Science
Mathematics
numerical linear algebra
communication avoiding algorithms
LU factorization
low-rank approximations
block low-rank matrices
triangular solve
data sparse matrices
linear systems
ISSN:0895-4798, 1095-7162
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Shrnutí:Block low-rank (BLR) compression can significantly reduce the memory and time costs of parallel sparse direct solvers. In this paper, we investigate the performance of the BLR triangular solve phase, which we observe to be underwhelming when dealing with many right-hand sides (RHS). We explain that this is because the bottleneck of the triangular solve is not in accessing the BLR LU factors, but rather in accessing the RHS, which are uncompressed. Motivated by this finding, we propose several new hybrid variants, which combine the right-looking and left-looking communication patterns to minimize the number of accesses to the RHS. We confirm via a theoretical analysis that these new variants can significantly reduce the total communication volume. We assess the impact of this reduction on the time performance on a range of real-life applications using the MUMPS solver, obtaining up to 20% time reduction.
ISSN:0895-4798
1095-7162
DOI:10.1137/23M1568600