Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods1

We analyze a possibility of turning off post-smoothing (relaxation) in geometric multigrid when used as a preconditioner in preconditioned conjugate gradient (PCG) 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 PCG and preconditioned steepest descent (PSD) methods. The eigen-value 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 tests show that removing the post-smoothing results in overall 40–50 percent acceleration, due to the high costs of smoothing and relatively insignificant decrease in convergence speed. We demonstrate that PSD-SMG and flexible PCG-SMG converge similarly if SMG post-smoothing is off. A theoretical justification is provided.