Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems

Computing ◽  
1996 ◽  
Vol 56 (3) ◽  
pp. 179-196 ◽  
Author(s):  
P. Vaněk ◽  
J. Mandel ◽  
M. Brezina
Keyword(s):  
Fourth Order ◽  
Elliptic Problems ◽  
Algebraic Multigrid ◽  
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 An improved convergence analysis of smoothed aggregation algebraic multigrid

2011 ◽  
Vol 19 (3) ◽  
pp. 441-469 ◽  
Author(s):  
Marian Brezina ◽  
Petr Vaněk ◽  
Panayot S. Vassilevski
Keyword(s):  
Convergence Analysis ◽  
Algebraic Multigrid ◽  
Smoothed Aggregation

Convergence of algebraic multigrid based on smoothed aggregation

2001 ◽  
Vol 88 (3) ◽  
pp. 559-579 ◽  
Author(s):  
Petr Van\vek ◽  
Marian Brezina ◽  
Jan Mandel
Keyword(s):  
Algebraic Multigrid ◽  
Smoothed Aggregation

Smoothed aggregation algebraic multigrid for stochastic PDE problems with layered materials

2014 ◽  
Vol 21 (2) ◽  
pp. 239-255 ◽  
Author(s):  
Marian Brezina ◽  
Alireza Doostan ◽  
Tom Manteuffel ◽  
Steve McCormick ◽  
John Ruge
Keyword(s):  
Layered Materials ◽  
Algebraic Multigrid ◽  
Stochastic Pde ◽  
Smoothed Aggregation
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