scholarly journals Analysis of an algebraic Petrov–Galerkin smoothed aggregation multigrid method

2008 ◽  
Vol 58 (12) ◽  
pp. 1861-1874 ◽  
Author(s):  
Hervé Guillard ◽  
Aleš Janka ◽  
Petr Vaněk
Keyword(s):  
Multigrid Method ◽  
Smoothed Aggregation

scholarly journals Algebraic multigrid methods

Acta Numerica ◽  
2017 ◽  
Vol 26 ◽  
pp. 591-721 ◽  
Author(s):  
Jinchao Xu ◽  
Ludmil Zikatanov
Keyword(s):  
Minimization Problem ◽  
Large Scale ◽  
Multigrid Method ◽  
Approximate Solutions ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Unified Framework ◽  
Smoothed Aggregation ◽  
Algebraic Multigrid Methods ◽  
Coarse Space

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.

scholarly journals A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zhao-Li Shen ◽  
Ting-Zhu Huang ◽  
Bruno Carpentieri ◽  
Chun Wen
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Numerical Experiments ◽  
Multigrid Method ◽  
Cost Effective ◽  
Multigrid Methods ◽  
Stationary Distributions ◽  
Convergence Behavior ◽  
Aggregation Procedure ◽  
Smoothed Aggregation

Smoothed aggregation multigrid method is considered for computing stationary distributions of Markov chains. A judgement which determines whether to implement the whole aggregation procedure is proposed. Through this strategy, a large amount of time in the aggregation procedure is saved without affecting the convergence behavior. Besides this, we explain the shortage and irrationality of the Neighborhood-Based aggregation which is commonly used in multigrid methods. Then a modified version is presented to remedy and improve it. Numerical experiments on some typical Markov chain problems are reported to illustrate the performance of these methods.

A new smoothed aggregation multigrid method for anisotropic problems

2009 ◽  
Vol 16 (1) ◽  
pp. 19-37 ◽  
Author(s):  
Michael W. Gee ◽  
Jonathan J. Hu ◽  
Raymond S. Tuminaro
Keyword(s):  
Multigrid Method ◽  
Smoothed Aggregation ◽  
Anisotropic Problems

scholarly journals A Flexible, Parallel, Adaptive Geometric Multigrid Method for FEM

2021 ◽  
Vol 47 (1) ◽  
pp. 1-27
Author(s):  
Thomas C. Clevenger ◽  
Timo Heister ◽  
Guido Kanschat ◽  
Martin Kronbichler
Keyword(s):  
Multigrid Method ◽  
Geometric Multigrid Method ◽  
Geometric Multigrid
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