This paper explores the application of kernel learning methods for parameter prediction and evaluation in the Algebraic Multigrid Method (AMG), focusing on several Partial Differential Equation (PDE) problems. AMG is an efficient iterative solver for large-scale sparse linear systems, particularly those derived from elliptic and parabolic PDE discretizations. However, its performance heavily relies on numerous parameters, which are often set empirically and are highly sensitive to AMG's effectiveness. Traditional parameter optimization methods are either computationally expensive or lack theoretical support. To address this, we propose a Gaussian Process Regression (GPR)-based strategy to optimize AMG parameters and introduce evaluation metrics to assess their effectiveness. Trained on small-scale datasets, GPR predicts nearly optimal parameters, bypassing the time-consuming parameter sweeping process. We also use kernel learning techniques to build a kernel function library and determine the optimal kernel function through linear combination, enhancing prediction accuracy. In numerical experiments, we tested typical PDEs such as the constant-coefficient Poisson equation, variable-coefficient Poisson equation, diffusion equation, and Helmholtz equation. Results show that GPR-predicted parameters match grid search results in iteration counts while significantly reducing computational time. A comprehensive analysis using metrics like mean squared error, prediction interval coverage, and Bayesian information criterion confirms GPR's efficiency and reliability. These findings validate GPR's effectiveness in AMG parameter optimization and provide theoretical support for AMG's practical application.

翻译：本文探讨了核学习方法在代数多重网格法（AMG）参数预测与评估中的应用，聚焦于若干偏微分方程（PDE）问题。AMG是求解大规模稀疏线性系统的高效迭代解法器，尤其适用于椭圆型和抛物型PDE离散化生成的系统。然而，其性能高度依赖于众多参数，这些参数通常依赖经验设置，且对AMG效能极为敏感。传统参数优化方法或计算成本高昂，或缺乏理论支撑。为此，我们提出一种基于高斯过程回归（GPR）的策略来优化AMG参数，并引入评估指标以衡量其有效性。GPR通过小规模数据集训练，可预测接近最优的参数，从而规避耗时的参数扫描过程。我们还利用核学习技术构建核函数库，并通过线性组合确定最优核函数，以提升预测精度。在数值实验中，我们测试了常系数泊松方程、变系数泊松方程、扩散方程及亥姆霍兹方程等典型PDE。结果表明，GPR预测的参数在迭代次数上与网格搜索结果相当，同时显著减少了计算时间。通过均方误差、预测区间覆盖率和贝叶斯信息准则等指标的综合分析，证实了GPR的高效性与可靠性。这些发现验证了GPR在AMG参数优化中的有效性，并为AMG的实际应用提供了理论支持。