StructMG: A Fast and Scalable Structured Algebraic Multigrid

TL;DR

StructMG introduces a fast, scalable structured algebraic multigrid method that automatically constructs hierarchical grids, reducing complexity and improving performance for large-scale sparse linear systems across various scientific applications.

Contribution

The paper presents StructMG, a novel multigrid solver with automatic grid construction, stencil-based coarsening, and a unified parallel framework, achieving superior speed and scalability over existing methods.

Findings

Achieves up to 15.5x speedup over hypre's multigrid preconditioners.

Demonstrates excellent scalability on ARM and X86 platforms.

Effectively solves large-scale problems in diverse scientific fields.

Abstract

Parallel multigrid is widely used as preconditioners in solving large-scale sparse linear systems. However, the current multigrid library still needs more satisfactory performance for structured grid problems regarding speed and scalability. Based on the classical 'multigrid seesaw', we derive three necessary principles for an efficient structured multigrid, which instructs our design and implementation of StructMG, a fast and scalable algebraic multigrid that constructs hierarchical grids automatically. As a preconditioner, StructMG can achieve both low cost per iteration and good convergence when solving large-scale linear systems with iterative methods in parallel. A stencil-based triple-matrix product via symbolic derivation and code generation is proposed for multi-dimensional Galerkin coarsening to reduce grid complexity, operator complexity, and implementation effort. A unified parallel framework of sparse triangular solver is presented to achieve fast convergence and high parallel efficiency for smoothers, including dependence-preserving Gauss-Seidel and incomplete LU methods. Idealized and real-world problems from radiation hydrodynamics, petroleum reservoir simulation, numerical weather prediction, and solid mechanics, are evaluated on ARM and X86 platforms to show StructMG's effectiveness. In comparison to \textit{hypre}'s structured and general multigrid preconditioners, StructMG achieves the fastest time-to-solutions in all cases with average speedups of 15.5x, 5.5x, 6.7x, 7.3x over SMG, PFMG, SysPFMG, and BoomerAMG, respectively. StructMG also significantly improves strong and weak scaling efficiencies.