A Survey of the Parallelization Issues for Geometric and Algebraic Multigrid Methods based on Generic Banded Approximate Inverses

2013 ◽  
Vol 7 ◽  
pp. 65-98
Author(s):  
G.A. Gravvanis ◽  
C.K. Filelis-Papadopoulos ◽  
P.I. Matskanidis
Keyword(s):  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Approximate Inverses ◽  
Algebraic Multigrid Methods

scholarly journals Algebraic multigrid methods for saddle point systems arising from mortar contact formulations

2021 ◽  
Author(s):  
T. A. Wiesner ◽  
M. Mayr ◽  
A. Popp ◽  
M. W. Gee ◽  
W. A. Wall
Keyword(s):  
Saddle Point ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Algebraic Multigrid Methods

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.

Cache Issues of Algebraic Multigrid Methods for Linear Systems with Multiple Right-Hand Sides

2005 ◽  
Vol 27 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Gundolf Haase ◽  
Stefan Reitzinger
Keyword(s):  
Linear Systems ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Right Hand ◽  
Algebraic Multigrid Methods

scholarly journals On the Ideal Interpolation Operator in Algebraic Multigrid Methods

2018 ◽  
Vol 56 (3) ◽  
pp. 1693-1710 ◽  
Author(s):  
Xuefeng Xu ◽  
Chen-Song Zhang
Keyword(s):  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Interpolation Operator ◽  
Ideal Interpolation ◽  
Algebraic Multigrid Methods ◽  
The Ideal

scholarly journals Comparison of geometric and algebraic multigrid methods in edge-based finite-element analysis

2005 ◽  
Vol 41 (5) ◽  
pp. 1672-1675 ◽  
Author(s):  
K. Watanabe ◽  
H. Igarashi ◽  
T. Honma
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Element Analysis ◽  
Edge Based ◽  
Algebraic Multigrid Methods

On algebraic multigrid methods derived from partition of unity nodal prolongators

2006 ◽  
Vol 13 (2-3) ◽  
pp. 105-131 ◽  
Author(s):  
Tim Boonen ◽  
Geoffrey Deliége ◽  
Stefan Vandewalle
Keyword(s):  
Partition Of Unity ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Algebraic Multigrid Methods

scholarly journals Algebraic Theory of Two-Grid Methods

2015 ◽  
Vol 8 (2) ◽  
pp. 168-198 ◽  
Author(s):  
Yvan Notay
Keyword(s):  
Algebraic Theory ◽  
Decomposition Methods ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Convergence Theory ◽  
Eigenvalue Bounds ◽  
Straightforward Application ◽  
Convergence Estimates ◽  
Nonsymmetric Matrices ◽  
Algebraic Multigrid Methods

AbstractAbout thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.

Algebraic Multigrid Methods (AMG) for the Efficient Solution of Fully Implicit Formulations in Reservoir Simulation

2007 ◽  
Author(s):  
Klaus Stueben ◽  
Tanja Clees ◽  
Hector Klie ◽  
Bo Lu ◽  
Mary Fanett Wheeler
Keyword(s):  
Reservoir Simulation ◽  
Efficient Solution ◽  
Multigrid Methods ◽  
Algebraic Multigrid ◽  
Fully Implicit ◽  
Algebraic Multigrid Methods
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