Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning

06/03/2021
by   Ali Taghibakhshi, et al.
0

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 paper, we consider the complementary problem of learning to coarsen graphs for a multigrid solver. We propose a method using a reinforcement learning (RL) agent based on graph neural networks (GNNs), which 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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

03/12/2020

Learning Algebraic Multigrid Using Graph Neural Networks

Efficient numerical solvers for sparse linear systems are crucial in sci...
02/02/2021

Graph Coarsening with Neural Networks

As large-scale graphs become increasingly more prevalent, it poses signi...
04/17/2022

Convergence analysis of a two-grid method for nonsymmetric positive definite problems

Multigrid is a powerful solver for large-scale linear systems arising fr...
06/16/2020

Multipole Graph Neural Operator for Parametric Partial Differential Equations

One of the main challenges in using deep learning-based methods for simu...
05/27/2021

Coarse-Grid Selection Using Simulated Annealing

Multilevel techniques are efficient approaches for solving the large lin...
11/07/2021

Convergence analysis of two-level methods with general coarse solvers

Multilevel methods are among the most efficient numerical methods for so...
11/19/2020

GL-Coarsener: A Graph representation learning framework to construct coarse grid hierarchy for AMG solvers

In many numerical schemes, the computational complexity scales non-linea...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.