Numerical Analysis and Scientific Computing Seminar

Nonsymmetric Preconditioning for Symmetric Linear Equations and Eigenvalue Problems

Speaker: Andrew Knyazev, Mitsubishi Electric Research Laboratories

Location: Warren Weaver Hall 1302

Date: Nov. 8, 2013, 10 a.m.

Synopsis:

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. 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 dramatically slows down the standard PCG. For the 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 converges nearly identical to flexible PCG if SMG post-smoothing is off. A theoretical justification is provided. [See {http://arxiv.org/abs/1212.6680}]