Algebraic Multigrid and the Fast Adaptive Composite Grid Method in Large Scale Computation.

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.

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Asynchronous Fast Adaptive Composite-Grid Methods for Elliptic Problems: Theoretical Foundations

SIAM Journal on Numerical Analysis ◽

10.1137/s0036142902400767 ◽

2004 ◽

Vol 42 (1) ◽

pp. 130-152 ◽

Cited By ~ 11

Author(s):

Barry Lee ◽

Stephen F. McCormick ◽

Bobby Philip ◽

Daniel J. Quinlan

Keyword(s):

Elliptic Problems ◽

Composite Grid ◽

Adaptive Composite ◽

Theoretical Foundations

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The MPS Reconstruction of Porous Media Using Multiple-Grid Templates

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.462-463.462 ◽

2013 ◽

Vol 462-463 ◽

pp. 462-465 ◽

Cited By ~ 1

Author(s):

Yi Du ◽

Ting Zhang

Keyword(s):

Porous Media ◽

Large Scale ◽

Grid Method ◽

Training Image ◽

Multiple Point ◽

Grid Data ◽

Large Scale Structures ◽

Complex Features ◽

Grid Nodes ◽

Data Template

It is difficult to reconstruct the unknown information only by some sparse known data in the reconstruction of porous media. Multiple-point geostatistics (MPS) has been proved to be a powerful tool to capture curvilinear structures or complex features in training images. One solution to capture large-scale structures while considering a data template with a reasonably small number of grid nodes is provided by the multiple-grid method. This method consists in scanning a training image using increasingly finer multiple-grid data templates instead of a big and dense data template. The experimental results demonstrate that multiple-grid data templates and MPS are practical in porous media reconstruction.

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