Algebraic multigrid methods

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.

A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

Smoothed aggregation multigrid method is considered for computing stationary distributions of Markov chains. A judgement which determines whether to implement the whole aggregation procedure is proposed. Through this strategy, a large amount of time in the aggregation procedure is saved without affecting the convergence behavior. Besides this, we explain the shortage and irrationality of the Neighborhood-Based aggregation which is commonly used in multigrid methods. Then a modified version is presented to remedy and improve it. Numerical experiments on some typical Markov chain problems are reported to illustrate the performance of these methods.

UA-AMG Methods for 2-D 1-T Radiation Diffusion Equations and Their CPU-GPU Implementations

In this paper, we study several unsmoothed aggregation based algebraic multigrid (UA-AMG) methods with regard to different characteristics of CPUs and graphics processing units (GPUs). We propose some UA-AMG methods with lower computational complexity for CPU and CPU-GPU, and study these UA-AMG methods mixing with 4 kinds of red-black colored Gauss-Seidel smoothers for CPU-GPU since the initial mesh is structured. These UA-AMG methods are used as preconditioners for the conjugate gradient (CG) solver to solve a class of two-dimensional single-temperature radiation diffusion equations discretized by preserving symmetry finite volume element scheme. Numerical results demonstrate that, UA-NA-CG-s, which wins the best robustness and efficiency among them, is much more efficient than the default AMG preconditioned CG solvers in HYPRE, AGMG and Cusp for CPU; Under CPU-GPU, UA-W-CG-p is the most robust and efficient one, and rather more efficient than the smoothed aggregation based AMG preconditioned CG solver in Cusp.