论文标题
最佳多项式Smoother用于Multigrid V-Cycles
Optimal polynomial smoothers for multigrid V-cycles
论文作者
论文摘要
重新审视了使用多项式方法在多族方法中改善简单平滑的迭代的想法，以进行对称正定（SPD）系统。当单步柔滑的本身对应于SPD操作员时，特别是一个非常简单的迭代，是Chebyshev半词的紧密表亲，基于第四类的Chebyshev多项式，而不是第一类，它优化了两级边界，可以返回Hackbusch。使用McCormick的V-Cycle理论得出了一般多项式SmoOther的完整V-Cycle结合。对于V-Cycle Bound的第四种Chebyshev迭代是准最佳的。可以从数值上找到V-Cycle结合的最佳多项式，与第四种Chebyshev迭代相比，误差收缩因子比第四次误差率低约18％，因为平滑步骤的数量转到无限。讨论了优化迭代的实现，并用一个简单的数值示例说明了多项式SmoOther的性能。
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.
