TY - JOUR
T1 - Algebraic multigrid methods
AU - Xu, Jinchao
AU - Zikatanov, Ludmil
N1 - Generated from Scopus record by KAUST IRTS on 2023-02-15
PY - 2017/5/1
Y1 - 2017/5/1
N2 - 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 nc is the span of the eigenvectors corresponding to the first nc eigenvectors RA (with R; = R + RT - RTAR). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with RA, 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.
AB - 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 nc is the span of the eigenvectors corresponding to the first nc eigenvectors RA (with R; = R + RT - RTAR). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with RA, 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.
UR - https://www.cambridge.org/core/product/identifier/S0962492917000083/type/journal_article
UR - http://www.scopus.com/inward/record.url?scp=85020450077&partnerID=8YFLogxK
U2 - 10.1017/S0962492917000083
DO - 10.1017/S0962492917000083
M3 - Article
SN - 1474-0508
VL - 26
SP - 591
EP - 721
JO - Acta Numerica
JF - Acta Numerica
ER -