A two‐grid SA‐AMG convergence bound that improves when increasing the polynomial degree

In this paper, we consider the convergence rate of a smoothed aggregation algebraic multigrid method, which uses a simple polynomial (1 ? t)^ν or an optimal Chebyshev‐like polynomial to construct the smoother and prolongation operator. The result is purely algebraic, whereas a required main weak approximation property of the tentative interpolation operator is verified for a spectral element agglomeration version of the method. More specifically, we prove that, for partial differential equations (PDEs), the two‐grid method converges uniformly without any regularity assumptions. Moreover, the convergence rate improves uniformly when the degree of the polynomials used for the smoother and the prolongation increases. Such a result, as is well‐known, would imply uniform convergence of the multilevel W‐cycle version of the algorithm. Numerical results, for both PDE and non‐PDE (graph Laplacian) problems are presented to illustrate the theoretical findings. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.     

An improved convergence analysis of smoothed aggregation algebraic multigrid

We present an improved analysis of the smoothed aggregation algebraic multigrid method extending the original proof in [Numer. Math. 2001; 88 :559–579] and its modification in [Multilevel Block Factorization Preconditioners. Matrix‐based Analysis and Algorithms for Solving Finite Element Equations. Springer: New York, 2008]. The new result imposes fewer restrictions on the aggregates that makes it easier to verify in practice. Also, we extend a result in [Appl. Math. 2011] that allows us to use aggressive coarsening at all levels. This is due to the properties of the special polynomial smoother that we use and analyze. In particular, we obtain bounds in the multilevel convergence estimates that are independent of the coarsening ratio. Numerical illustration is also provided. Copyright © 2011 John Wiley & Sons, Ltd.     

Smoothed aggregation multigrid for incompressible flows

We discuss the advantages of using algebraic multigrid based on smoothed aggregation for solving indefinite linear problems. The ingredients of smoothed aggregation are used to construct a black-box monolithic multigrid method with indefinite coarse problems. Although we discuss some techniques for enforcing the uniform inf-sup stability on all coarse levels, numerical experiments suggest that it is not strictly necessary. The proposed multigrid preconditioner shows robust behaviour for different time-step parameters and even for very elongated geometries, where other techniques based on h -independent preconditioners of the pressure Schur complement lose their efficiency. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

We analyze a general multigrid method with aggressive coarsening and polynomial smoothing. We use a special polynomial smoother that originates in the context of the smoothed aggregation method. Assuming the degree of the smoothing polynomial is, on each level k, at least Ch _k+1/h _k , we prove a convergence result independent of h _k+1/h _k . The suggested smoother is cheaper than the overlapping Schwarz method that allows to prove the same result. Moreover, unlike in the case of the overlapping Schwarz method, analysis of our smoother is completely algebraic and independent of geometry of the problem and prolongators (the geometry of coarse spaces).     

Reducing complexity of algebraic multigrid by aggregation

A typical approach to decrease computational costs and memory requirements of classical algebraic multigrid methods is to replace a conservative coarsening algorithm and short‐distance interpolation on a fixed number of fine levels by an aggressive coarsening with a long‐distance interpolation. Although the quality of the resulting algebraic multigrid grid preconditioner often deteriorates in terms of convergence rates and iteration counts of the preconditioned iterative solver, the overall performance can improve substantially. We investigate here, as an alternative, a possibility to replace the classical aggressive coarsening by aggregation, which is motivated by the fact that the convergence of aggregation methods can be independent of the problem size provided that the number of levels is fixed. The relative simplicity of aggregation can lead to improved solution and setup costs. The numerical experiments show the relevance of the proposed combination on both academic and benchmark problems in reservoir simulation from oil industry. Copyright © 2016 John Wiley & Sons, Ltd.     

Algebraic analysis of aggregation‐based multigrid

A convergence analysis of two‐grid methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M‐)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter, which in some sense measures its quality. The procedure is purely algebraic and can be used to control a posteriori the quality of automatic coarsening algorithms. Assuming the aggregation pattern is sufficiently regular, it is further shown that the resulting bound is asymptotically sharp for a large class of elliptic boundary value problems, including problems with variable and discontinuous coefficients. In particular, the analysis of typical examples shows that the convergence rate is insensitive to discontinuities under some reasonable assumptions on the aggregation scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Algebraic interface‐based coarsening AMG preconditioner for multi‐scale sparse matrices with applications to radiation hydrodynamics computation

Coarsening is a crucial component of algebraic multigrid (AMG) methods for iteratively solving sparse linear systems arising from scientific and engineering applications. Its application largely determines the complexity of the AMG iteration operator. Usually, high operator complexities lead to fast convergence of the AMG method; however, they require additional memory and as such do not scale as well in parallel computation. In contrast, although low operator complexities improve parallel scalability, they often lead to deterioration in convergence. This study introduces a new type of coarsening strategy called algebraic interface‐based coarsening that yields a better balance between convergence and complexity for a class of multi‐scale sparse matrices. Numerical results for various model‐type problems and a radiation hydrodynamics practical application are provided to show the effectiveness of the proposed AMG solver.     

Multigrid for the Galerkin least squares method in linear elasticity

In SIAM J. Numer. Anal. 28 (1991) 1680-1697, Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid method. This multigrid is robust in that the convergence is uniform as the parameter ν goes to 1/2. Computational experiments are included.     

Notes on convergence of an algebraic multigrid method

The convergence theory for algebraic multigrid (AMG) algorithms proposed in Chang and Huang [Q.S. Chang, Z.H. Huang, Efficient algebraic multigrid algorithms and their convergence, SIAM J. Sci. Comput. 24 (2002) 597–618] is further discussed and a smaller and elegant upper bound is obtained. On the basis of element-free AMGe [V.E. Henson, P.S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23(2) (2001) 629–650] we rewrite the interpolation operator for the classical AMG (cAMG), present a uniform expression and then, by introducing a sparse approximate inverse in the Frobenius norm, give a general convergence theorem which is suited for not only cAMG but also AMG for finite elements and element-free AMGe.     

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.     

Algebraic multigrid and 4th‐order discrete‐difference equations of incompressible fluid flow

This paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th‐order discrete‐difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream‐function field). One is based on a classical, algebraic multigrid, method (C‐AMG) the other is based on a smoothed‐aggregation method for 4th‐order problems (SA‐AMG). In the C‐AMG case, the inter‐grid transfer operators are enhanced using Jacobi relaxation. In the SA‐AMG case, they are improved using a constrained energy optimization of the coarse‐grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA‐AMG approach, however, is more costly both in storage and operations. The Jacobi‐relaxed C‐AMG approach is faster, by a factor of between 2 and 4 for two‐dimensional problems, even though its reduction factors are inferior to those of SA‐AMG. For non‐uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd.     

Convergence analysis of multigrid methods with residual scaling techniques

In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper.     

Improved convergence bounds for smoothed aggregation method: Linear dependence of the convergence rate on the number of levels

The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear prolongator smoother. The tentative prolongator is responsible for the approximation, while the prolongator smoother enforces the smoothness of the coarse-level basis functions.     

Multigrid methods for the symmetric interior penalty method on graded meshes

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi‐optimal error estimates in both the energy norm and the L₂ norm for the SIP method, and prove uniform convergence of the W‐cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd.     

An algebraic multigrid method for finite element discretizations with edge elements

This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H₀(curl,Ω). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl‐operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

WEB-Spline–Based Multigrid Methods for the Stationary Stokes Problem

We discuss a multigrid technique in solving a large system of linear algebraic equations arising in the approximation of Stokes equations by a new strategy based on weighted extended B-spline (WEB-spline) methods. Three types of WEB-spline–based Stokes elements satisfying the inf-sup condition are considered. First for a linear-constant type of Stokes element, we give the detailed multigrid algorithm and its convergence proof. The convergence proof of the multigrid algorithm for a bubble-stabilized WEB-spline–based Stokes element is dealt with separately. Multigrid method in the case of bubble-condensed variational form is simplified using the techniques from the bubble-stabilized case.     

A new smoothed aggregation multigrid method for anisotropic problems

A new prolongator is proposed for smoothed aggregation (SA) multigrid. The proposed prolongator addresses a limitation of standard SA when it is applied to anisotropic problems. For anisotropic problems, it is fairly standard to generate small aggregates (used to mimic semi‐coarsening) in order to coarsen only in directions of strong coupling. Although beneficial to convergence, this can lead to a prohibitively large number of non‐zeros in the standard SA prolongator and the corresponding coarse discretization operator. To avoid this, the new prolongator modifies the standard prolongator by shifting support (non‐zeros within a prolongator column) from one aggregate to another to satisfy a specified non‐zero pattern. This leads to a sparser operator that can be used effectively within a multigrid V‐cycle. The key to this algorithm is that it preserves certain null space interpolation properties that are central to SA for both scalar and systems of partial differential equations (PDEs). We present two‐dimensional and three‐dimensional numerical experiments to demonstrate that the new method is competitive with standard SA for scalar problems, and significantly better for problems arising from PDE systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Algebraic analysis of two‐grid methods: The nonsymmetric case

Two‐grid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are provided that characterize all eigenvalues of the iteration matrix. In the symmetric positive‐definite (SPD) case, these expressions reproduce the sharp two‐grid convergence estimate obtained by Falgout, Vassilevski and Zikatanov (Numer. Linear Algebra Appl. 2005; 12 :471–494), and also previous algebraic bounds, which can be seen as corollaries of this estimate. These results allow to measure the convergence by checking ‘approximation properties’. In this work, proper extensions of the latter to the nonsymmetric case are presented. Sometimes approximation properties for the SPD case are summarized in loose terms; e.g.: Interpolation must be able to approximate an eigenvector with error bound proportional to the size of the eigenvalue (SIAM J. Sci. Comp. 2000; 22 :1570–1592). It is shown that this can be applied to nonsymmetric problems too, understanding ‘size’ as ‘modulus’. Eventually, an analysis is developed, for the nonsymmetric case, of the theoretical foundations of ‘compatible relaxation’, according to which a Fine/Coarse partitioning may be checked and possibly improved. Copyright © 2009 John Wiley & Sons, Ltd.