AMG based on compatible weighted matching for GPUs

TL;DR

This paper presents an efficient GPU implementation of an algebraic multigrid preconditioner based on a novel coarsening technique using compatible weighted matching, outperforming existing GPU preconditioners in solving PDE-related linear systems.

Contribution

It introduces a GPU-tailored AMG method utilizing a new coarsening approach based on compatible weighted matching and parallel maximum weight matching algorithms.

Findings

Outperforms Nvidia AmgX preconditioners in GPU environments

Provides high-quality coarse matrices for PDE discretizations

Efficiently exploits GPU parallelism in all computational kernels

Abstract

We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic Multigrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPU. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (spd) matrices, named "coarsening based on compatible weighted matching". It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred as the Suitor algorithm, and show that it allow us to obtain good quality coarse matrices for our AMG on GPUs. We exploit inherent parallelism of modern GPUs in all the kernels involving sparse matrix computations both for the setup of the preconditioner and for its application in a Krylov solver, outperforming preconditioners available in Nvidia AmgX library. We report results about a large set of linear systems arising from discretization of scalar and vector partial differential equations (PDEs).