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30
A review of algebraic multigrid
, 2001
"... Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to b ..."
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Cited by 347 (11 self)
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Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrixbased approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes, both in 2D and 3D. Since AMG does not make use of any geometric information, it is a “plugin ” solver which can even be applied to problems without any geometric background, provided that the
Convergence of Algebraic Multigrid Based on Smoothed Aggregation
 COMPUTING
, 1998
"... We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, ..."
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Cited by 125 (14 self)
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We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be apriori verified computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements.
Robustness and scalability of algebraic multigrid
 SIAM J. SCI. COMPUT
, 1998
"... Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a ..."
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Cited by 52 (9 self)
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Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we demonstrate that range of applicability, while describing some of the recent advances in AMG technology. Moreover, in light of the imperatives of modern computer environments, we also examine AMG in terms of algorithmic scalability. Finally, we show some of the situations in which standard AMG does not work well, and indicate the current directions taken by AMG researchers to alleviate these difficulties.
A ParticlePartition Of Unity Method For The Solution Of Elliptic, Parabolic And Hyperbolic PDEs
 SIAM J. SCI. COMP
"... In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used a ..."
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Cited by 35 (6 self)
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In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h or pversion. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convectiondiffusion, instationary diffusion, linear advection and elliptic problems.
Flow Field Clustering via Algebraic Multigrid
 In Proc. IEEE Visualization Conf. ’04
, 2004
"... We present a novel multiscale approach for flow visualization. We define a local alignment tensor that encodes a measure for alignment to the direction of a given flow field. This tensor induces an anisotropic differential operator on the flow domain, which is discretized with a standard finite elem ..."
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Cited by 27 (4 self)
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We present a novel multiscale approach for flow visualization. We define a local alignment tensor that encodes a measure for alignment to the direction of a given flow field. This tensor induces an anisotropic differential operator on the flow domain, which is discretized with a standard finite element technique. The entries of the corresponding stiffness matrix represent the anisotropically weighted couplings of adjacent nodes of the domain mesh. We use an algebraic multigrid algorithm to generate a hierarchy of fine to coarse descriptions for the above coupling data. This hierarchy comprises a set of coarse grid nodes, a multiscale of basis functions and their corresponding supports. We use these supports to obtain a multilevel decomposition of the flow structure. Standard streamline icons are used to visualize this decomposition at any userselected level of detail. The method provides a single framework for vector field decomposition independent on the domain dimension or mesh type. Applications are shown in 2D, for flow fields on curved surfaces, and for 3D volumetric flow fields. 1
An Algebraic Multigrid Method for Linear Elasticity
 SIAM J. Sci. Comp
, 2003
"... Abstract. We present an algebraic multigrid (AMG) method for the efficient solution of linear (block)systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise f ..."
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Cited by 21 (3 self)
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Abstract. We present an algebraic multigrid (AMG) method for the efficient solution of linear (block)systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the blockinterpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is wellknown from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio ν. We obtain rates ρ < 0.4 for ν < 0.4. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.
Analysis Of Tensor Product Multigrid
 Numer. Algorithms
, 1999
"... We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other dir ..."
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Cited by 16 (1 self)
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We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced. The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semicoarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at onedimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid Vcycle are bounded independently of the number of grid le...
Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses
, 2000
"... Sparse approximate inverses are considered as smoothers for multigrid. They are based on the SPAIAlgorithm (Grote and Huckle, 1997), which constructs a sparse approximate inverse M of a matrix A by minimizing I MA in the Frobenius norm. This yields a new hierarchy of smoothers: SPAI0, SPAI1, ..."
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Cited by 14 (4 self)
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Sparse approximate inverses are considered as smoothers for multigrid. They are based on the SPAIAlgorithm (Grote and Huckle, 1997), which constructs a sparse approximate inverse M of a matrix A by minimizing I MA in the Frobenius norm. This yields a new hierarchy of smoothers: SPAI0, SPAI1, SPAI("). Advantages of SPAI smoothers over classical smoothers are inherent parallelism, possible local adaptivity and improved robustness. The simplest smoother, SPAI0, is based on a diagonal matrix M . It is shown to satisfy the smoothing property for symmetric positive denite problems. Numerical experiments show that SPAI0 smoothing is usually preferable to damped Jacobi smoothing. In more dicult situations, where the simpler SPAI0 and SPAI1 smoothers are not adequate, the SPAI(") smoother provides a natural procedure for improvement where needed. Numerical examples illustrate the usefulness of SPAI smoothing.
Spacetime approximation with sparse grids
"... In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to socalled spacetime sparse grid spaces. For a uniform discretization of the spa ..."
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Cited by 13 (1 self)
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In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to socalled spacetime sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(N d) degrees of freedom, these spaces involve for d> 1 also only O(N d) degrees of freedom for the discretization of the whole spacetime problem. But they provide the same approximation rate as classical spacetime Finite Element spaces which need O(N d+1) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and for timedependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid spacetime spaces for general stable multiscale bases. We then restrict ourselves to an interpolatory multiscale basis, i.e. a hierarchical basis. Here, to be able to handle also complicated spatial domains Ω, we construct the hierarchical basis from a given spatial Finite Element basis as follows: First we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method. Then, we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach. This way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our spacetime sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various spacetime problems with two spatial dimensions. Also implementational issues, data structures and questions of adaptivity are addressed to some extent.