TY - JOUR

T1 - Algebraic multigrid methods

AU - Xu, Jinchao

AU - Zikatanov, Ludmil

N1 - Funding Information:
The work of the first author was supported by US DOE grant DE-SC-0009249, as part of the 'Collaboratory on Mathematics for Mesoscopic Modeling of Materials', and NSF grants DMS-1412005 and DMS-1522615. The work of the second author was supported by NSF grants DMS-1418843 and DMS-1522615.
Publisher Copyright:
© 2017 Cambridge University Press.

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.

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U2 - 10.1017/S0962492917000083

DO - 10.1017/S0962492917000083

M3 - Article

AN - SCOPUS:85020450077

SN - 0962-4929

VL - 26

SP - 591

EP - 721

JO - Acta Numerica

JF - Acta Numerica

ER -