Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches

TL;DR

This paper introduces a machine learning approach to compute simpler, spectrally equivalent coarse-grid operators in algebraic multigrid methods, aiming to reduce operator complexity while preserving convergence for parametric PDEs.

Contribution

It develops novel neural network algorithms guided by AMG theory to effectively reduce coarse-grid operator complexity in AMG methods.

Findings

ML algorithms successfully reduce operator complexity

Maintains AMG convergence for parametric PDEs

Outperforms existing methods in numerical experiments

Abstract

We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods, addressing the well-known issue of increasing operator complexity. Guided by the AMG theory on spectrally equivalent coarse-grid operators, we have developed novel ML algorithms that utilize neural networks (NNs) combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall AMG convergence for solving parametric partial differential equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and compare with existing methods for computing non-Galerkin coarse-grid operators.