Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches
This addresses a well-known bottleneck in numerical solvers for partial differential equations, offering a domain-specific incremental improvement.
The paper tackles the problem of increasing operator complexity in algebraic multigrid (AMG) methods by proposing a data-driven machine learning approach to compute non-Galerkin coarse-grid operators, resulting in reduced complexity while maintaining convergence for parametric PDE problems.
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.