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.     

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.     

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.     

Analysis of an aggregation‐based algebraic two‐grid method for a rotated anisotropic diffusion problem

A two‐grid convergence analysis based on the paper [Algebraic analysis of aggregation‐based multigrid, by A. Napov and Y. Notay, Numer. Lin. Alg. Appl. 18 (2011), pp. 539–564] is derived for various aggregation schemes applied to a finite element discretization of a rotated anisotropic diffusion equation. As expected, it is shown that the best aggregation scheme is one in which aggregates are aligned with the anisotropy. In practice, however, this is not what automatic aggregation procedures do. We suggest approaches for determining appropriate aggregates based on eigenvectors associated with small eigenvalues of a block splitting matrix or based on minimizing a quantity related to the spectral radius of the iteration matrix. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient algebraic multigrid method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions

The quality of the mesh used in the finite element discretizations will affect the efficiency of solving the discreted linear systems. The usual algebraic solvers except multigrid method do not consider the effect of the grid geometry and the mesh quality on their convergence rates. In this paper, we consider the hierarchical quadratic discretizations of three‐dimensional linear elasticity problems on some anisotropic hexahedral meshes and present a new two‐level method, which is weakly independent of the size of the resulting problems by using a special local block Gauss–Seidel smoother, that is LBGS_v iteration when used for vertex nodes or LBGS_m iteration for midside nodes. Moreover, we obtain the efficient algebraic multigrid (AMG) methods by applying DAMG (AMG based on distance matrix) or DAMG‐PCG (PCG with DAMG as a preconditioner) to the solution of the coarse level equation. The resulting AMG methods are then applied to a practical example as a long beam. The numerical results verify the efficiency and robustness of the proposed AMG algorithms. Copyright © 2015 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.     

Generalizing reduction-based algebraic multigrid

Algebraic multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.     

An algebraic multigrid method for elasticity based on an auxiliary topology with edge matrices

This article introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second-order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two-level convergence even for systems of equations such as linear elasticity . This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.     

An algebraic multigrid method for finite element systems on criss-cross grids

In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid method. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday The work was supported in part by NSAF(10376031) and National Major Key Project for basic researches and by National High-Tech ICF Committee in China.     

Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids

The purpose of this work is to reduce the CPU time necessary to solve three two-dimensional linear diffusive problems governed by Laplace and Poisson equations, discretized with anisotropic grids. The Finite Difference Method is used to discretizate the differential equations with central differencing scheme. The systems of equations are solved with the lexicographic and red–black Gauss–Seidel methods associated to the geometric multigrid with correction scheme and V-cycle. The anisotropic grids considered have aspect ratios varying from 1/1024 up to 16,384. Four algorithms are compared: full coarsening, semicoarsening, full coarsening followed by semicoarsening and partial semicoarsening. Three new restriction schemes for anisotropic grids are proposed: geometric half weighting, geometric full weighting and partial weighting. Comparisons are made among these three new schemes and some restriction schemes presented in literature: injection, half weighting and full weighting. The prolongation process used is the bilinear interpolation. It is also investigated the effects on the CPU time caused by: the number of inner iterations of the smoother, the number of grids and the number of grid elements. It was verified that the partial semicoarsening algorithm is the fastest. This work also provides the optimum values of the multigrid components for this algorithm.     

A new cascadic multigrid

We present a new cascadic multigrid for elliptic problems.     

A Rayleigh quotient minimization algorithm based on algebraic multigrid

This paper presents a new algebraic extension of the Rayleigh quotient multigrid (RQMG) minimization algorithm to compute the smallest eigenpairs of a symmetric positive definite pencil ( A , M ). Earlier versions of RQMG minimize the Rayleigh quotient over a hierarchy of geometric grids. We replace the geometric mesh information with the algebraic information defined by an algebraic multigrid preconditioner. At each level, we minimize the Rayleigh quotient with a block preconditioned algorithm. Numerical experiments illustrate the efficiency of this new algorithm to compute several eigenpairs. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonnested multigrid methods for linear problems

This article presents an application of nonnested and unstructured multigrid methods to linear elastic problems. A variational formulation for transfer operators and some multigrid strategies, including adaptive algorithms, are presented. Expressions for the performance evaluation of multigrid strategies and its comparison with direct and preconditioned conjugate gradient algorithms are also presented. A C++ implementation of the multigrid algorithms and the quadtree and octree data structures for transfer operators are discussed. Some two‐ and three‐dimensional elasticity examples are analyzed. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:313–331, 2001     

V-cycle convergence of some multigrid methods for ill-posed problems

For ill-posed linear operator equations we consider some V-cycle multigrid approaches, that, in the framework of Bramble, Pasciak, Wang, and Xu (1991), we prove to yield level independent contraction factor estimates. Consequently, we can incorporate these multigrid operators in a full multigrid method, that, together with a discrepancy principle, is shown to act as an iterative regularization method for the underlying infinite-dimensional ill-posed problem. Numerical experiments illustrate the theoretical results.

     

A note on the upper bound in SA AMG convergence analysis

In [M. Brezina, P. Vaněk and P. S. Vassilevski, An improved convergence of smoothed aggregation algebraic multigrid, Numer. Linear Algebra Appl., 19 (2012), pp. 441–469], a uniform convergence bound for smoothed aggregation algebraic multigrid with aggressive coarsening and massive polynomial prolongator and multigrid smoothers is established provided that the number of smoothing steps is equal to the coarsening ratio parameter ν. The final convergence estimate needs the uniform bound for the constant C_ν ∕ (2ν + 1). In this note, we give an improved upper bound. Copyright © 2013 John Wiley & Sons, Ltd.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

On some versions of the element agglomeration AMGe method

The present paper deals with element‐based algebraic multigrid (AMGe) methods that target linear systems of equations coming from finite element discretizations of elliptic partial differential equations. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We study a number of variants of the spectral agglomerate AMGe method. The core of the algorithms relies on element agglomeration utilizing the element topology (built recursively from fine to coarse levels). The actual selection of the coarse degrees of freedom is based on solving a large number of local eigenvalue problems. Additionally, we investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V‐cycle utilizing simple interpolation matrices and nested conjugate gradient (CG)‐based recursive calls between the levels. The presented algorithms are illustrated with an extensive set of experiments based on a matlab implementation of the methods. Copyright © 2008 John Wiley & Sons, Ltd.     

OpenMG: A new multigrid implementation in Python

In many large‐scale computations, systems of equations arise in the form Au = b, where A is a linear operation to be performed on the unknown data u, producing the known right‐hand side, b, which represents some constraint of known or assumed behavior of the system being modeled. Because such systems can be very large, solving them directly can be too slow. In contrast, a multigrid method removes different components of the error at different resolutions using smoothers that reduce high‐frequency components of the error more readily than low. Here, we present an open‐source multigrid solver written only in Python. OpenMG is a pure Python experimentation environment for testing multigrid concepts, not a production solver. The particular restriction method implemented is for ‘standard’ multigrid. By making the code simple and modular, we make the algorithmic details clear. The resulting solver is tested on an implicit pressure reservoir simulation problem with satisfactory results.Copyright © 2013 John Wiley & Sons, Ltd.     

A new method based on SOM network to generate coarse meshes for overlapping unstructured multigrid algorithm

A new method to generate coarse meshes for overlapping unstructured multigrid algorithm based on self-organizing map (SOM) neural network is presented in this paper. The application of SOM neural network can overcome some limitations of conventional methods and which is designed to pursuit the best structure relation between fine and coarse unstructured meshes with the object to ensure robust convergence for overlapping unstructured multigrid algorithm. Besides, this method can automate the generation of unstructured meshes and is suitable for both two and three dimensions conditions.