scholarly journals Convergence in Norm of Nonsymmetric Algebraic Multigrid

2019 ◽  
Vol 41 (5) ◽  
pp. S269-S296 ◽  
Author(s):  
Tom Manteuffel ◽  
Ben S. Southworth
Keyword(s):  
Algebraic Multigrid ◽  
Convergence In Norm

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 Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem

2013 ◽  
Vol 35 (5) ◽  
pp. S3-S17 ◽  
Author(s):  
Panayot S. Vassilevski ◽  
Umberto Villa
Keyword(s):  
Algebraic Multigrid ◽  
Block Diagonal

GPU Acceleration of Algebraic Multigrid Preconditioners for Discrete Elliptic Field Problems

2014 ◽  
Vol 50 (2) ◽  
pp. 461-464 ◽  
Author(s):  
Christian Richter ◽  
Sebastian Schops ◽  
Markus Clemens
Keyword(s):  
Algebraic Multigrid ◽  
Gpu Acceleration ◽  
Multigrid Preconditioners
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