Bringing physics into the coarse‐grid selection: Approximate diffusion distance/effective resistance measures for network analysis and algebraic multigrid for graph Laplacians and systems of elliptic partial differential equations
Bringing physics into the coarse‐grid selection: Approximate diffusion distance/effective resistance measures for network analysis and algebraic multigrid for graph Laplacians and systems of elliptic partial differential equations
复制标题
将物理学引入粗网格选择：用于网络分析的近似扩散距离/有效阻力测量以及用于图拉普拉斯算子和椭圆偏微分方程组的代数多重网格
DOI：
10.1002/nla.2539
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发表时间：
2023
影响因子：
4.3
通讯作者：
Lee, Barry
中科院分区：
文献类型：
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作者：
Lee, Barry
在最近的一篇论文中，作者研究了在椭圆型偏微分方程（PDEs）系统中选择粗自由度（CDOFs）或粗节点（C节点）的关联亲和度量。该方法应用于一组松弛向量，暴露了PDE算子的近零空间分量。利用这种亲和度度量选择cdof，并利用最小二乘法构造插值算子，提出了一种代数多重网格（AMG）方法。然而，这个AMG求解器有几个值得注意的问题。首先，为了捕获强各向异性，可能需要大量的测试向量；其次，求解器的性能对随机测试向量的初始集很敏感。这两个问题都反映了该措施的敏感统计性质。在本文中，我们推导了其他几个统计度量，这些度量可以改善这些问题并提高AMG性能。这些措施与马尔可夫过程有关，PDE本身可以对其建模。具体来说，这些措施是基于扩散距离/有效阻力的过程，因此，这些措施纳入物理到CDOF的选择。此外，由于扩散距离/有效阻力可用于分析图网络，这些措施也为分析大规模网络提供了一种非常经济的方案。本文给出了这些度量的推导，并给出了在加权图拉普拉斯算子和椭圆型边值问题系统上分析网络和AMG性能的数值实验。
In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented.