Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration

TL;DR

This paper enhances the AMLI-cycle method by simplifying convergence theory using Chebyshev polynomials and introducing a momentum-accelerated variant that is easy to implement and maintains robust, efficient performance.

Contribution

It revisits AMLI-cycle with Chebyshev polynomials for simplified convergence analysis and proposes a momentum-accelerated AMLI-cycle that avoids eigenvalue estimation, improving practicality.

Findings

The Chebyshev-based AMLI-cycle achieves uniform convergence without eigenvalue estimation.

The momentum-accelerated AMLI-cycle maintains uniform condition number and is as effective as Chebyshev-based methods.

Numerical experiments demonstrate the robustness and efficiency of the proposed momentum-accelerated AMLI-cycle.

Abstract

In this paper, we investigate the AMLI-cycle method and make two contributions. First, we revisit the AMLI-cycle using the Chebyshev polynomials and establish a theory for its uniform convergence, assuming the two-grid method converges uniformly. This removes the need for estimating extreme eigenvalues at all coarse levels. Only an estimation of the two-grid convergence rate is needed, which could be done on the second coarsest level, simplifying implementation and reducing computational costs for large-scale problems. Second, we introduce a momentum-accelerated AMLI-cycle using polynomials from momentum accelerations. This novel approach ensures a uniform condition number without requiring extreme eigenvalue or two-grid convergence rate estimations, making its implementation as straightforward as standard multigrid methods. We prove that it is asymptotically as good as the AMLI-cycle using the Chebyshev polynomials when the quadratic momentum-accelerated polynomials is used. Numerical experiments confirm the robustness and efficiency of the momentum-accelerated AMLI-cycle across various problems, demonstrating performance comparable to the Chebyshev-based AMLI-cycle. These findings validate the theoretical advantages and practical efficacy of the momentum-accelerated AMLI-cycle.