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.

翻译：代数多重网格法（AMG）是求解大型稀疏线性方程组 $Ax=b$ 最常用的迭代算法之一。在AMG中，粗网格是影响算法效率的关键组成部分，其构造依赖于强阈值参数 $\theta$。该参数通常凭经验选取，当前多数AMG求解器中默认值为：二维问题取0.25，三维问题取0.5。然而在许多实际问题中，粗网格质量与AMG算法效率对 $\theta$ 高度敏感；默认值很少是最优的，有时甚至远非如此。因此，如何选择更优的 $\theta$ 值是一个重要问题。本文针对实际问题中广泛使用的多尺度稀疏线性方程组，提出了一种基于深度学习的自动调优方法AutoAMG($\theta$)。该方法采用图神经网络（GNNs）提取矩阵特征，并通过多层感知机（MLP）建立矩阵特征与最优 $\theta$ 之间的映射关系，能够针对不同矩阵自适应输出 $\theta$ 值。数值实验表明，与默认 $\theta$ 值相比，AutoAMG($\theta$) 可实现显著的加速效果。