scholarly journals Multigrid methods for block‐Toeplitz linear systems: convergence analysis and applications

2020 ◽  
Author(s):  
Marco Donatelli ◽  
Paola Ferrari ◽  
Isabella Furci ◽  
Stefano Serra‐Capizzano ◽  
Debora Sesana
Keyword(s):  
Convergence Analysis ◽  
Linear Systems ◽  
Multigrid Methods

Multigrid Methods for Systems of Hyperbolic Conservation Laws

2013 ◽  
Vol 11 (2) ◽  
pp. 586-614 ◽  
Author(s):  
Swathi Amarala ◽  
Justin W. L. Wan
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Multigrid Methods ◽  
Hyperbolic Conservation

Load Balancing for Adaptive Multigrid Methods

1998 ◽  
Vol 19 (4) ◽  
pp. 1303-1321 ◽  
Author(s):  
Peter Bastian
Keyword(s):  
Load Balancing ◽  
Multigrid Methods

B‐Spline‐Based Monotone Multigrid Methods

2007 ◽  
Vol 45 (3) ◽  
pp. 1175-1199 ◽  
Author(s):  
Markus Holtz ◽  
Angela Kunoth
Keyword(s):  
Multigrid Methods ◽  
B Spline

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.

Nonlinear multigrid methods for total variation image denoising

2004 ◽  
Vol 7 (3-4) ◽  
pp. 199-206 ◽  
Author(s):  
Claudia Frohn-Schauf ◽  
Stefan Henn ◽  
Kristian Witsch
Keyword(s):  
Image Denoising ◽  
Total Variation ◽  
Multigrid Methods ◽  
Nonlinear Multigrid ◽  
Nonlinear Multigrid Methods
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