Compatible Relaxation and Coarsening in Algebraic Multigrid

Brannick, James; Falgout, Robert D.

doi:10.1137/090772216

Cited by 54 publications

(93 citation statements)

References 16 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Because of this focus, we obviate the problem of choosing a good coarse grid by considering cases where this is known in advance. In general, selection of the coarse grid could be obtained by a compatible relaxation process; see . The study of relaxation‐corrected bootstrap algebraic multigrid ( r BAMG) here is an attempt to systematically analyze the behavior of the algorithm in terms relative to several parameters.…”

Section: Introductionmentioning

confidence: 99%

Relaxation‐corrected bootstrap algebraic multigrid (rBAMG)

Brezina

Ketelsen

Manteuffel

et al. 2012

Numerical Linear Algebra App

View full text Add to dashboard Cite

SUMMARY Bootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an improved form of BAMG, called relaxation‐corrected bootstrap algebraic multigrid (rBAMG), that involves adding scaled residuals of the test vectors to the least‐squares equations. The basic rBAMG scheme was introduced in an earlier paper [1] and analyzed on a simple model problem. The purpose of the current paper is to further develop this algorithm by incorporating several new critical components and to systematically study its performance on an interesting model problem from quantum chromodynamics. Whereas the earlier paper introduced a new least‐squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed here to accurately estimate the convergence factors of the evolving algebraic multigrid solver. Such a capability is essential to the effective development of a fast solver, and the approach introduced here is shown numerically to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process developed here is an adaptive cycling process. This component assesses the effectiveness of the V‐cycle constructed in the initial rBAMG phase by applying it to the homogeneous equation. When poor convergence is observed, the set of test vectors is enhanced with the resulting error, enabling the subsequent least‐squares fit of interpolation to produce an improved V‐cycle. A related component is the scaling and recombination Ritz process that targets the so‐called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces. The aim of the numerical study documented here is to provide insight into the various design choices that arise in the development of an rBAMG algorithm. With this in mind, the results for quantum chromodynamics focus on the behavior of rBAMG in terms of the number of initial test vectors used, the number of relaxation sweeps applied to them, and the size of the target matrices. Copyright © 2012 John Wiley & Sons, Ltd.

show abstract

Section: Introductionmentioning

confidence: 99%

Relaxation‐corrected bootstrap algebraic multigrid (rBAMG)

Brezina

Ketelsen

Manteuffel

et al. 2012

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…It gives a practical way to measure the quality of a set of coarse variables, indeed, since in an efficient multigrid method relaxation scheme has to be effective on the fine variables, the convergence rate of a compatible relaxation scheme can be used as a measure of the quality of a set of coarse variables. This basic idea was used in different approaches to select coarse grids [6,15]. Here, we apply the principle of compatible relaxation to choose the best coarse matrix from the two available matrices in (3.2), and the corresponding coarse index set, by applying a simple point-wise relaxation scheme to the homogeneous systems associated to each of the matrices, starting from a random initial guess and then relaxing on the two complementary vector spaces separately.…”

Section: Main Ingredients For Coarseningmentioning

confidence: 99%

“…The above iterates provide approximation to the lowest eigenmode of B −1 A, which is commonly referred to as algebraic smooth vectors with respect to the current AMG method. If the convergence factor of the method is close to one, we can select w = x k / x k A and apply one of the coarsening algorithms described in the preceding section to generate a new method B 1 based on this new vector w. Assuming that we have constructed two (or more) methods B r , r = 0, 1, ..., m via the above bootstrap scheme aimed at improving the initial AMG, 6 we consider the homogeneous system and monitor the convergence of the following composite method, starting with a random initial guess x 0 ,…”

Section: Composite Amg With Prescribed Convergence Ratementioning

confidence: 99%

Adaptive AMG with coarsening based on compatible weighted matching

D'Ambra

Vassilevski

2013

Comput. Visual Sci.

View full text Add to dashboard Cite

We introduce a new composite adaptive Algebraic Multigrid (composite αAMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of αAMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on maximum weighted matching in a graph and on principles of compatible relaxation. The latter replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes.

show abstract

“…and discretized on unstructured grids with the aFEM finite element package, which has been used previously in [28,29,14]. The resulting matrices have some positive off-diagonal entries and are not diagonally dominant.…”

Section: Problem Descriptionsmentioning

confidence: 99%

Multigrid Smoothers for Ultraparallel Computing

Baker¹,

Falgout²,

Kolev³

et al. 2011

SIAM J. Sci. Comput.

Self Cite

141

114

View full text Add to dashboard Cite

Abstract. This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running on parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [22,23] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several C-F , polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultra-parallel computers. We also introduce the more robust 1 smoothers, which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems [29].

show abstract

Compatible Relaxation and Coarsening in Algebraic Multigrid

Cited by 54 publications

References 16 publications

Relaxation‐corrected bootstrap algebraic multigrid (rBAMG)

Relaxation‐corrected bootstrap algebraic multigrid (rBAMG)

Adaptive AMG with coarsening based on compatible weighted matching

Multigrid Smoothers for Ultraparallel Computing

Contact Info

Product

Resources

About