Optimal polynomial smoothers for multigrid V-cycles

TL;DR

This paper explores the use of optimized polynomial smoothers, especially Chebyshev-based methods, to enhance multigrid V-cycle efficiency for SPD systems, achieving significant error reduction.

Contribution

It introduces a new class of polynomial smoothers, including a fourth-kind Chebyshev iteration, and derives bounds for their performance within multigrid V-cycles.

Findings

Fourth-kind Chebyshev iteration is quasi-optimal for V-cycle bounds.

Numerical optimization yields polynomials with 18% lower error contraction.

Implementation details and numerical examples demonstrate effectiveness.

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 V-cycle 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 goes to infinity. Implementation of the optimized iteration is discussed, and the performance of the polynomial smoothers are illustrated with a simple numerical example.