Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation
Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation
复制标题
通过多级紧缩加速亥姆霍兹方程的移位拉普拉斯预条件子
DOI：
10.1016/j.jcp.2016.06.025
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
C. Vuik
中科院分区：
文献类型：
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作者：
A. H. Sheikh;D. Lahaye;L. G. Ramos;R. Nabben;C. Vuik
许多重要的物理现象都可以用亥姆霍兹方程来描述。我们调查在何种程度上的亥姆霍兹方程的移位拉普拉斯预条件的收敛可以加速使用多重网格矢量通缩。因此，我们提出了一个统一的框架，两个公布的算法。第一种方法使预处理操作器放气，无需进一步预处理。第二个收缩的原始运营商，并结合紧缩和预处理乘法的方式。我们追求两个科学贡献。首先，我们表明，使用模型问题分析，这两种算法集群的特征值。这里新的和关键的见解是，粗网格算子的近核导致一组有限的本征值以与波数成比例的距离远离集群的中心。这种效果在第一算法变体中不太明显，代价是更高的计算成本。在第二个贡献中，我们第一次量化了特征值聚类和迭代次数减少所导致的CPU时间的大量减少。为此，我们报告的结果，在PETSc的实施上的二维和三维的问题与恒定和可变波数。
Many important physical phenomena can be described by the Helmholtz equation. We investigate to what extent the convergence of the shifted Laplacian preconditioner for the Helmholtz equation can be accelerated using deflation with multigrid vectors. We therefore present a unified framework for two published algorithms. The first deflates the preconditioned operator and requires no further preconditioning. The second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion. We pursue two scientific contributions. First we show, using a model problem analysis, that both algorithms cluster the eigenvalues. The new and key insight here is that the near-kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number. This effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost. In the second contribution we quantify for the first time the large amount of reduction in CPU-time that results from the clustering of eigenvalues and the reduction in iteration count. We report to this end on the findings of an implementation in PETSc on two and three-dimensional problems with constant and variable wave number.