Optimization-based algebraic multigrid coarsening using reinforcement learning

Large sparse linear systems of equations are ubiquitous in science and engineering, such as those arising from discretizations of partial differential equations. Algebraic multigrid (AMG) methods are one of the most common methods of solving such linear systems, with an extensive body of underlying mathematical theory. A system of linear equations defines a graph on the set of unknowns and each level of a multigrid solver requires the selection of an appropriate coarse graph along with restriction and interpolation operators that map to and from the coarse representation. The efficiency of the multigrid solver depends critically on this selection and many selection methods have been developed over the years. Recently, it has been demonstrated that it is possible to directly learn the AMG interpolation and restriction operators, given a coarse graph selection. In this dissertation, we consider the complementary problem of learning to coarsen graphs for a multigrid solver. We study a method using a reinforcement learning (RL) agent based on two different neural network type; convolutional neural network (CNN) and graph neural networks (GNNs). The agent with CNN architecture is trained and tested on structured grids; however, the agent with GNN architecture can learn to perform graph coarsening on small training graphs and then be applied to unstructured large graphs. We demonstrate that this method can produce better coarse graphs than existing algorithms, even as the graph size increases and other properties of the graph are varied. We also propose an efficient inference procedure for performing graph coarsening that results in linear time complexity in graph size.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Taghibakhshi, Ali

Contributors dc:contributor

• West, Mathew

• Reinforcement Learning
• Multigrid
• Graph Neural Network

Statement dc:rights

• Copyright 2021 Ali Taghibakhshi

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113063

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F113063