Reducing Parallel Communication in Algebraic Multigrid through Sparsification
- Univ. of Illinois at Urbana-Champaign, IL (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
Algebraic multigrid (AMG) is an $$\mathcal{O}(n)$$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids which utilize complementary relaxation and interpolation operators is constructed. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grid matrices and reduced there. However, large parallel communication costs often limit parallel scalability. Furthermore, as the multigrid hierarchy is formed, each coarse matrix is formed through a triple matrix product. The resulting coarse grids often have significantly more nonzeros per row than the original fine-grid operator, thereby generating high parallel communication costs associated with sparse matrix-vector multiplication (SpMV) on coarse levels. In this paper, we introduce a method that systematically removes entries in coarse-grid matrices after the hierarchy is formed, leading to improved communication costs. We sparsify by removing weakly connected or unimportant entries in the matrix, leading to improved solve time. The main trade-off is that if the heuristic identifying unimportant entries is used too aggressively, then AMG convergence can suffer. To counteract this, the original hierarchy is retained, allowing entries to be reintroduced into the solver hierarchy if convergence is too slow. This enables a balance between communication cost and convergence, as necessary. In this paper we present new algorithms for reducing communication and present a number of computational experiments in support.
- Research Organization:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- AC52-07NA27344
- OSTI ID:
- 1808755
- Report Number(s):
- LLNL-JRNL--673388; 796162
- Journal Information:
- SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 38; ISSN 1064-8275
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
Preparing sparse solvers for exascale computing
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journal | January 2020 |
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