Reducing Parallel Communication in Algebraic Multigrid through Sparsification

TL;DR

This paper proposes a sparsification method to reduce parallel communication costs in algebraic multigrid by removing weakly connected entries from coarse-grid matrices, balancing efficiency and convergence.

Contribution

It introduces a systematic sparsification technique for coarse-grid matrices in AMG, improving scalability without significantly harming convergence.

Findings

Reduced communication costs in AMG hierarchies.

Maintained convergence by reintroducing entries if needed.

Faster solve times demonstrated in computational experiments.

Abstract

Algebraic multigrid (AMG) is an $\mathcal{O}(n)$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids is constructed that utilize complementary relaxation and interpolation operators. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grids and reduced there. However, large parallel communication costs often limit parallel scalability. 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 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 an 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.