Asset Details
Do you wish to reserve the book?
Algebraic multigrid methods
Hey, we have placed the reservation for you!
By the way, why not check out events that you can attend while you pick your title.
Oops! Something went wrong.
Looks like we were not able to place the reservation. Kindly try again later.
Are you sure you want to remove the book from the shelf?
Algebraic multigrid methods
Do you wish to request the book?
Algebraic multigrid methods
Please be aware that the book you have requested cannot be checked out. If you would like to checkout this book, you can reserve another copy
We have requested the book for you!
Your request is successful and it will be processed during the Library working hours. Please check the status of your request in My Requests.
Oops! Something went wrong.
Looks like we were not able to place your request. Kindly try again later.
Journal Article
Algebraic multigrid methods
2017
Overview
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.
Publisher
Cambridge University Press
Subject
/ Methods
