This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother$R$for a matrix$A$, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension$n_{c}$is the span of the eigenvectors corresponding to the first$n_{c}$eigenvectors$\bar{R}A$(with$\bar{R}=R+R^{T}-R^{T}AR$). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with$\bar{R}A$, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.

中文翻译：

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代数多重网格方法

本文概述了求解大规模方程组的 AMG 方法，例如偏微分方程的离散化方法。AMG通常被理解为'algebraic multigrid'的首字母缩写，但也可以理解为'abstract multigrid'。实际上，我们在本文中演示了如何以及为什么可以在更抽象的层次上更好地理解代数多重网格方法。在文献中，已经从不同的角度开发了许多不同的代数多重网格方法。在本文中，我们尝试开发一个统一的框架和理论，可以用于以连贯的方式推导和分析不同的代数多重网格方法。给定一个更平滑的$R$对于矩阵$澳元，例如 Gauss-Seidel 或 Jacobi，我们证明了维数的最优粗空间$n_{c}$是对应于第一个特征向量的跨度$n_{c}$特征向量$\bar{R}澳元（和$\bar{R}=R+R^{T}-R^{T}AR$）。我们还证明了这个最优的粗糙空间可以通过一个与$\bar{R}澳元，并证明大多数现有 AMG 方法的粗空间可以看作是这个迹最小化问题的近似解。此外，我们提供了构建准最优粗空间的一般方法，并且我们证明了在适当的假设下，得到的底层线性系统的两级 AMG 方法关于问题的大小一致收敛，系数变化和各向异性。我们的理论适用于大多数现有的多重网格方法，包括标准几何多重网格方法、经典 AMG、能量最小化 AMG、非平滑和平滑聚合 AMG 和谱 AMGe。