AutoAMG($\theta$): An Auto-tuned AMG Method Based on Deep Learning for Strong Threshold

TL;DR

This paper introduces AutoAMG(θ), a deep learning-based auto-tuning approach for the strong threshold parameter in Algebraic Multigrid methods, improving efficiency by adaptively selecting optimal θ values for different matrices.

Contribution

It proposes a novel GNN and MLP-based framework to automatically tune the strong threshold parameter in AMG, enhancing performance over default settings.

Findings

AutoAMG(θ) achieves significant speedup over default θ values.

The method adaptively selects better θ for diverse matrices.

Numerical experiments validate the effectiveness of the approach.

Abstract

Algebraic Multigrid (AMG) is one of the most used iterative algorithms for solving large sparse linear equations $Ax=b$. In AMG, the coarse grid is a key component that affects the efficiency of the algorithm, the construction of which relies on the strong threshold parameter $\theta$. This parameter is generally chosen empirically, with a default value in many current AMG solvers of 0.25 for 2D problems and 0.5 for 3D problems. However, for many practical problems, the quality of the coarse grid and the efficiency of the AMG algorithm are sensitive to $\theta$; the default value is rarely optimal, and sometimes is far from it. Therefore, how to choose a better $\theta$ is an important question. In this paper, we propose a deep learning based auto-tuning method, AutoAMG($\theta$) for multiscale sparse linear equations, which are widely used in practical problems. The method uses Graph Neural Networks (GNNs) to extract matrix features, and a Multilayer Perceptron (MLP) to build the mapping between matrix features and the optimal $\theta$, which can adaptively output $\theta$ values for different matrices. Numerical experiments show that AutoAMG($\theta$) can achieve significant speedup compared to the default $\theta$ value.