论文标题
复兴的AMLI循环：从Chebyshev多项式到动量加速
Rejuvenating AMLI-Cycle: From Chebyshev Polynomials to Momentum Acceleration
论文作者
论文摘要
在本文中，我们研究了AMLI-Cycle方法，并做出了两种贡献。首先，我们使用Chebyshev多项式重新访问AMLI周期，并为其均匀收敛建立理论，假设两网格方法均匀收敛。这消除了在所有粗级别上估计极端特征值的必要性。仅需要估计两网格收敛速率，这可以在第二个粗糙的水平上进行，从而简化实施并降低大规模问题的计算成本。其次，我们使用动量加速度的多项式引入了动量加速的AMLI周期。这种新颖的方法确保了统一的条件数，而无需极端特征值或两个网格的收敛率估计，从而使其实现与标准的多机方法一样简单。我们证明，当使用二次动量加速多项式时，使用Chebyshev多项式渐近地与AMLI周期一样好。数值实验证实了各种问题上动量加速AMLI循环的鲁棒性和效率，表明性能与基于Chebyshev的AMLI-Cycle相当。这些发现证明了动量加速AMLI周期的理论优势和实际功效。
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.
