Kernel-learning parameter prediction and evaluation in algebraic multigrid method for several PDEs

TL;DR

The paper tackles AMG parameter tuning by deploying a kernel-learning Gaussian Process Regression framework that predicts near-optimal smoothing/coarsening parameters from small-scale training data. It builds a kernel-function library and learns combinations to tailor the GP kernel, with hyperparameters selected via likelihood optimization across PDE-driven datasets. A rich evaluation suite combines $R^2$, Corr, $PICP$, $MSE$, $RMSE$, $MAE$, $BIC$, MdAPE, and LOO-SPE to assess accuracy and reliability. Numerical experiments on constant and variable coefficient Poisson, diffusion, and Helmholtz equations show that GPR-predicted AMG parameters reproduce iteration counts close to grid search while dramatically reducing setup time, validating the approach's practicality and reliability. The work suggests a scalable, theoretically grounded path to robust AMG deployment and points to promising extensions with mixed precision, complex PDEs, and online learning for dynamic adaptation.

Abstract

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.

Figures (6)

• Figure 1: The variation of the number of iterations for AMG solving the constant coefficient Poisson equation with different connectivity parameters $\theta$ when $n$ is 64, 256, and 400, respectively
• Figure 2: Regression curves for predicting $\theta$ with respect to $n$ using GPR for AMG solving the constant coefficient Poisson equation
• Figure 3: When $T=2$, the computational domain is uniformly partitioned into four blocks $(B_{i}, i=1,2,3,4)$. Diffusion coefficient $\kappa$ is the same in each block, when $B_{i}\neq B_{j}$, $k_{i}\neq k_{j}$.
• Figure 4: Traversal plots for three sets of test matrices, along with iteration steps corresponding to GPR prediction, default values, and optimal parameters
• Figure 5: Variation of the number of iterations for AMG solving the Helmholtz equation with coefficient $2\pi$ and different connectivity parameters $\theta$ when $n$ is 64,256,400, respectively

Theorems & Definitions (1)

• Definition 2.1