A fast algebraic multigrid preconditioned conjugate gradient solver

doi:10.1016/j.amc.2005.11.115

Applied Mathematics and Computation

Volume 179, Issue 1, 1 August 2006, Pages 344-351

https://doi.org/10.1016/j.amc.2005.11.115 Get rights and content

Abstract

This work presents a new approach for selecting the coarse grids allowing a faster algebraic multigrid (AMG) preconditioned conjugate gradient solver. This approach is based on an appropriate choice of the parameter α considering the matrix density during the coarsening process which implies in a significant reduction in the matrix dimension at all AMG levels.

Introduction

The multigrid method (MG) is a well-established numerical technique for solving linear systems. Several works have explored the use of MG as a preconditioner for the conjugate gradient method (CG) [1], [2], [3]. In [2] the MG is used with the CG in the resolution of the two-dimensional Poisson equation in a regular domain, showing the superiority of this method in relation to the incomplete Cholesky conjugate gradient method (ICCG).

In contrast to MG, where a mesh hierarchy is explicitly required, AMG constructs the matrix hierarchy and transfer operators just using information from the original matrix. Hence, when the problem involves an irregular domain and unstructured meshes or when one is interested on black-box solvers the AMG is well suited as a preconditioner for iterative solvers.

The standard AMG normally presents a high computational cost, thus it is not generally used in small and medium size problems [4]. In [1] a modification in the conventional AMG is proposed to reduce its construction and solving time. In that work the original matrix is approximated to a symmetric M-matrix. However, when the original matrix is very different of a M-matrix that approach can increase prohibitively the convergence factor.

In this work, we present a new approach to select the coarse grids that allows the use of the AMG as a preconditioner for the CG method for small and medium problems. This approach leads to a significant reduction of the matrix dimension at all levels without losing the robustness.

The paper is organized as follows. In Section 2 we give a brief overview of the basic AMG algorithm, detailing the coarsening process in Section 3. In Sections 4 Test cases, 5 Results, the test cases and the numerical results are presented, comparing the performance of the AMG with that of incomplete Cholesky method in the preconditioning of the ill-conditioned matrices. Finally, some conclusions are formulated in Section 6.

Section snippets

The algebraic multigrid

In this section we give an outline of the basic principles of AMG, and define some terminology and notation. Detailed explanations may be found in [3]. Consider a problem of the form $Au = f,$ where A is an n × n matrix with entries a_ij. For AMG, a “grid” is simply a set of indices of the variables, so the original grid is denoted by ω_k = {1, 2, … , n} with k = 1.

In any multigrid method, the central idea is that error e, that is not eliminated by relaxation, must be removed by coarse-grid correction. This is

The coarsening process

The AMG efficiency is improved by reducing the number of nonzero entries of the coarse matrices by using the following sets of connections between grid points [1], [3], [4], [5], [7]: $\begin{matrix} N_{i} = {j : | a_{ij} | \neq 0, i \neq j}, \\ S_{i} = {j \in N_{i} : | a_{ij} | ⩾ α \max_{k \neq i} | a_{ik} |}, \\ S_{i}^{T} = {j : i \in S_{j}}, \end{matrix}$ where N_i is the direct neighborhood of a point i, S_i is the set of the points that strongly influence the point i and $S_{i}^{T}$ is the points that are strongly influenced by point i. Here the influence concept is defined in terms of the absolute value of the

Test cases

The proposed AMG preconditioning method was applied in the resolution of linear systems with ill-conditioned matrices. These matrices were obtained from the Matrix Market [6] and Davis [12] sparse matrix collections and have the characteristics summarized in Table 1. In that table, the fourth column indicates the average nonzero per column and row in the matrix. All the matrices are symmetric and positive definite, hence the incomplete Cholesky conjugate gradient method (ICCG) was used for

Results

The convergence of iterative methods like CG can be understood in terms of the eigenvalue analysis of the preconditioned matrix [2]. Thus, we analyze the efficiency of preconditioner AMG through of eigenvalue distribution analysis of the matrix after the preconditioning.

Fig. 2 shows the eigenvalue distribution for matrix bcsstk14 after the applications of the AMG and the incomplete Cholesky method (IC). For AMG the eigenvalues are clustered around 1 and a few eigenvalues are scattered between 1

Conclusions

The paper presented a new approach to the use of the AMG as a preconditioner for the CG method in the resolution of small and medium size ill-conditioned problems. The proposed approach allowed a significant reduction in the AMG setup and solver times and the storage space required by the operators and for the right sides and approximation vectors over all grids of the AMG.

The results presented prove the robustness of the method for the test problems. For matrix bcsstk17, in which the use of

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A fast algebraic multigrid preconditioned conjugate gradient solver

Abstract

Introduction

Section snippets

The algebraic multigrid

The coarsening process

Test cases

Results

Conclusions

Comput. Methods Appl. Mech. Engrg.

J. Comput. Phys.

Appl. Math. Comput.

J. Comput. Appl. Math.

A Multigrid Tutorial