Learning to Solve Faster with Data-Driven Algebraic Multigrid Methods

Project Overview

Solving large sparse linear systems poses a significant computational bottleneck in numerous scientific and engineering applications relevant to LLNL's interests. Algebraic Multigrid (AMG) methods have long served as important tools, offering fast and scalable solutions for such problems on High-Performance Computing (HPC) systems. However, the development of robust and efficient AMG methods remains challenging, particularly for complex problems, and has become increasingly critical as the intricacies of underlying physics and computing platforms continue to evolve. In this project, we embarked on an exploration to assess the feasibility of leveraging Machine Learning (ML) approaches in conjunction with insights gleaned from classical sparse linear solvers. This work aimed to illuminate the path toward the creation of novel, ML-informed, and data-driven AMG algorithms. We demonstrated the viability of this approach through the application of Supervised Learning techniques using Neural Networks (NNs). These NNs have exhibited the capability to compute AMG smoothers and coarse-grid operators that are more efficient compared to conventional numerical methods. Furthermore, we have developed a Python-based codebase (https://github.com/LLNL/NISS) to facilitate ongoing research in this promising direction.

Mission Impact

This project carries substantial significance across multiple Lawrence Livermore National Laboratory (LLNL) missions. It supports High-Performance Computing, Simulation, and Data Science by advancing the understanding and development of sparse linear solvers for challenging problems of vital interest to LLNL in order to achieve high performance on HPC systems. It fosters the evolution of using ML techniques for scientific applications by developing the science and technology tools. It also supports High-Energy-Density Science, Nuclear, Chemical and Isotopic Science and Technology and the Weapon Simulation and Computing program, which rely heavily on simulations that employ scalable parallel linear solvers.

Publications, Presentations, and Patents

Kai Chang, Rui Peng Li, Ru Huang, and Yuanzhe Xi, "Reducing Operator Complexity in Algebraic Multigrid with Machine Learning" (Presentation, 21st Copper Mountain Conference On Multigrid Methods, Copper Mountain, CO, 2023).

Rui Peng Li, Yunhui He, "Data-driven Multigrid methods based on local Fourier analysis" (Presentation, 21st Copper Mountain Conference On Multigrid Methods, Copper Mountain, CO, 2023).