Nonsymmetric multigrid preconditioning for conjugate gradient methods

TL;DR

This paper investigates the impact of disabling post-smoothing in geometric multigrid preconditioning for conjugate gradient methods, demonstrating that it can accelerate convergence in certain scenarios without significant drawbacks.

Contribution

It provides a numerical and theoretical analysis showing that turning off post-smoothing can speed up multigrid preconditioned iterative methods for 3D Laplacian problems.

Findings

Post-smoothing can be avoided in flexible PCG and LOBPCG without loss of convergence speed.

Disabling post-smoothing accelerates the solver due to reduced computational costs.

The acceleration effect is independent of memory interconnection.

Abstract

We numerically analyze the possibility of turning off post-smoothing (relaxation) in geometric multigrid when used as a preconditioner in conjugate gradient linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening Multigrid (SMG) method is provided by the hypre parallel software package. We solve linear systems using two variants (standard and flexible) of the preconditioned conjugate gradient (PCG) and preconditioned steepest descent (PSD) methods. The eigenvalue problems are solved using the locally optimal block preconditioned conjugate gradient (LOBPCG) method available in hypre through BLOPEX software. We observe that turning off the post-smoothing in SMG dramatically slows down the standard PCG-SMG. For flexible PCG and LOBPCG, our numerical results show that post-smoothing can be avoided, resulting in overall acceleration, due to the high costs of smoothing and relatively insignificant decrease in convergence speed. We numerically demonstrate for linear systems that PSD-SMG and flexible PCG-SMG converge similarly if SMG post-smoothing is off. We experimentally show that the effect of acceleration is independent of memory interconnection. A theoretical justification is provided.