Optimal polynomial smoothers for multigrid V-cycles

Abstract

The idea of using polynomial methods to improve simple smoother iterations within a multigrid method for a symmetric positive definite (SPD) system is revisited. When the single-step smoother itself corresponds to an SPD operator, there is in particular a very simple iteration, a close cousin of the Chebyshev semi-iterative method, based on the Chebyshev polynomials of the fourth instead of first kind, that optimizes a two-level bound going back to Hackbusch. A full Vcycle bound for general polynomial smoothers is derived using the V-cycle theory of McCormick. The fourth-kind Chebyshev iteration is quasi-optimal for the V-cycle bound. The optimal polynomials for the V-cycle bound can be found numerically, achieving an about 18% lower error contraction factor bound than the fourth-kind Chebyshev iteration, asymptotically as the number of smoothing steps k → ∞. Implementation of the optimized iteration is discussed, and the performance of the polynomial smoothers are illustrated with a simple numerical example.(i.e., polynomial) acceleration techniques can be applied to improve the smoother iterations. General-purpose Krylov methods don't necessarily have good smoothing properties, since they are typically designed to optimize some norm of the total error and not just its "un-smooth" components. In particular, a method like conjugate gradients can dramatically worsen the performance of a smoother iteration on the "highfrequency" error components in return for a marginal improvement in the performance on the "smooth" error components, which should of course be left for the coarse grid correction to handle. There are, however, polynomial methods specifically designed to be used as smoothers. One popular strategy for symmetric positive definite (SPD) systems, suggested by Adams, Brezina, Hu, and Tamaro [1], is to use the standard Chebyshev semi-iteration, but adjusted to target only an upper portion [λ * , λ max ] of the range of eigenvalues [λ min , λ max ], with λ * an ad-hoc parameter. While this method works well in practice, and it is reported that the performance is not very sensitive to the particular choice of λ * , the presence of the ad-hoc parameter is still unaesthetic. We will see that we can design methods that avoid such ad-hoc parameters, by picking polynomials that optimize multigrid convergence bounds. We will restrict our attention to SPD systems and single-step SPD smoothers (since polynomial methods tend to be most effective in the SPD setting), and examine both a two-level bound for general polynomial smoothers going back to Hackbusch [4], as well as develop a full V-cycle convergence bound for general polynomial smoothers starting from the V-cycle theory developed by McCormick [7]. We will see that a very simple Chebyshev-type iteration, which avoids the need for any ad-hoc parameter, optimizes the two-level bound and is quasi-optimal for the V-cycle bound. We will also see that a small modification to the iteration, though somewhat complicated to implement, optimizes the V-cycle bound. The effe...

Optimized sparse approximate inverse smoothers for solving Laplacian linear systems

“…Though more effective than the weighted Jacobi smoother, the popular Gauss-Seidel smoother is not very friendly to massively parallel computers due to its sequential nature [1,38]. For general symmetric positive definite linear systems, significant efforts in the development of multigrid solvers have been concentrated on the design of effective parallelizable smoothers with smaller smoothing factors (and faster convergence rates), see for example [9,20,27,34,35,36,46] and the references therein. In [16], the authors compared three different Chebyshev polynomial smoothers in the context of aggressive coarsening, where the one-dimensional minimization formulations are defined over a finite interval that bounds all the eigenvalues of diagonally preconditioned system.…”