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145
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 276 (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 siz ..."
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Cited by 104 (12 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. Key words. Algebraic multigrid, zero energy modes, convergence theory, computational mechanics, Finite Elements, iterative solvers 1. Introduction. This paper is concerned with the analysis of an Algebraic Multigrid Method (AMG) based on smoothed aggregation, which we have introduced in [28], and which in turn is a further development of [25, 26]. This method and its ...
Algebraic Multigrid Based On Element Interpolation (AMGe)
, 1998
"... We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global meas ..."
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Cited by 87 (14 self)
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We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.
BoomerAMG: a Parallel Algebraic Multigrid Solver and Preconditioner
 Applied Numerical Mathematics
, 2000
"... Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sh ..."
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Cited by 82 (7 self)
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Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarsegrid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [7] for the selection of coarsegrid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this pa...
ACE: A Fast Multiscale Eigenvector Computation for Drawing Huge Graphs
, 2002
"... We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits ..."
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Cited by 64 (13 self)
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We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of a million nodes in less than a minute. Moreover, the algorithm can deal with more general entities, such as graphs with masses and negative weights (to be defined in the text), and it appears to be applicable outside of graph drawing too.
Texture segmentation by multiscale aggregation of filter responses and shape elements
 IN ICCV
, 2003
"... Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A Textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottomup aggregation framework that combines structural characteristics of te ..."
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Cited by 55 (8 self)
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Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A Textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottomup aggregation framework that combines structural characteristics of texture elements with filter responses. Our process adaptively identifies the shape of texture elements and characterize them by their size, aspect ratio, orientation, brightness, etc., and then uses various statistics of these properties to distinguish between different textures. At the same time our process uses the statistics of filter responses to characterize textures. In our process the shape measures and the filter responses crosstalk extensively. In addition, a topdown cleaning process is applied to avoid mixing the statistics of neighboring segments. We tested our algorithm on real images and demonstrate that it can accurately segment regions that contain challenging textures.
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
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Cited by 54 (15 self)
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Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
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 44 (8 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.
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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