Nonsymmetric Multigrid Preconditioning for Conjugate Gradient Methods

    •  Bouwmeester, H.; Dougherty, A.; Knyazev, A.V., "Nonsymmetric Multigrid Preconditioning for Conjugate Gradient Methods," Tech. Rep. TR2013-027, arXiv, June 2013.
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      • @techreport{Bouwmeester2013jun,
      • author = {Bouwmeester, H. and Dougherty, A. and Knyazev, A.V.},
      • title = {Nonsymmetric Multigrid Preconditioning for Conjugate Gradient Methods},
      • journal = {arXiv},
      • year = 2013,
      • month = jun,
      • url = {}
      • }
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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 exible) 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 exible 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 elatively insignificant decrease in convergence speed. We numerically demonstrate for linear systems that PSD-SMG and exible 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.