scholarly journals Compatible Relaxation and Coarsening in Algebraic Multigrid

2010 ◽  
Vol 32 (3) ◽  
pp. 1393-1416 ◽  
Author(s):  
James J. Brannick ◽  
Robert D. Falgout
Keyword(s):  
Algebraic Multigrid ◽  
Compatible Relaxation

scholarly journals Bootstrap Algebraic Multigrid: Status Report, Open Problems, and Outlook

2015 ◽  
Vol 8 (1) ◽  
pp. 112-135 ◽  
Author(s):  
Achi Brandt ◽  
James Brannick ◽  
Karsten Kahl ◽  
Ira Livshits
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Algebraic Multigrid ◽  
Status Report ◽  
Open Problems ◽  
Highly Oscillatory ◽  
Test Vectors ◽  
Development Analysis ◽  
Main Ideas ◽  
Compatible Relaxation

AbstractThis paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.

scholarly journals Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

2018 ◽  
Vol 40 (3) ◽  
pp. A1473-A1493 ◽  
Author(s):  
James Brannick ◽  
Fei Cao ◽  
Karsten Kahl ◽  
Robert D. Falgout ◽  
Xiaozhe Hu
Keyword(s):  
Algebraic Multigrid ◽  
Optimal Interpolation ◽  
Compatible Relaxation

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
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