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15
ANALYSIS OF MULTISCALE METHODS
, 2004
"... The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method. ..."
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Cited by 118 (13 self)
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The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method.
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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A Multidimensional Approach to ForceDirected Layouts of Large Graphs
, 2000
"... Abstract. We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or thr ..."
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Cited by 36 (5 self)
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Abstract. We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional drawings onto two or three dimensions often results in drawings that are “smoother ” and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a midrange PC. 1
A Fast MultiDimensional Algorithm for Drawing Large Graphs
 In Graph Drawing’00 Conference Proceedings
, 2000
"... We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensi ..."
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Cited by 28 (4 self)
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We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a midrange PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...
Multiscale modeling and computation
 Notices Amer. Math. Soc
, 2003
"... Multiscale modeling and computation is a rapidly evolving area of research that will have a fundamental impact on computational science and applied mathematics and will influence the way we view the relation between mathematics and science. Even though multiscale problems have long been studied in m ..."
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Cited by 15 (2 self)
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Multiscale modeling and computation is a rapidly evolving area of research that will have a fundamental impact on computational science and applied mathematics and will influence the way we view the relation between mathematics and science. Even though multiscale problems have long been studied in mathematics, the current excitement is driven mainly by the use of mathematical models in the applied sciences: in particular, material science, chemistry, fluid dynamics, and biology. Problems in these areas are often multiphysics in nature; namely, the processes at different scales are governed by physical laws of different character: for example, quantum mechanics at one scale and classical mechanics at another. Emerging from this intense activity is a need for new mathematics and new ways of interacting with mathematics. Fields such as mathematical physics and stochastic processes, which have so far remained in the background as far as modeling and computation is concerned, will move to the frontier. New questions will arise, new priorities will be set as a result of the rapid evolution in the computational fields. There are several reasons for the timing of the current interest. Modeling at the level of a single scale, such as molecular dynamics or continuum
Optimal Multigrid Algorithms for the Massive Gaussian Model and Path Integrals
, 1996
"... Multigrid algorithms are presented which, in addition to eliminating the critical slowing down, can also eliminate the "volume factor". The elimination of the volume factor removes the need to produce many independent finegrid configurations for averaging out their statistical deviations, by averag ..."
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Cited by 7 (6 self)
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Multigrid algorithms are presented which, in addition to eliminating the critical slowing down, can also eliminate the "volume factor". The elimination of the volume factor removes the need to produce many independent finegrid configurations for averaging out their statistical deviations, by averaging over the many samples produced on coarse grids during the multigrid cycle. Thermodynamic limits of observables can be calculated to relative accuracy " r in just O(" \Gamma2 r ) computer operations, where " r is the error relative to the standard deviation of the observable. In this paper, we describe in details the calculation of the susceptibility in the one dimensional massive Gaussian model, which is also a simple example of path integral. Numerical experiments show that the susceptibility can be calculated to relative accuracy " r in about 8" \Gamma2 r random number generations, independently of the mass size. KEY WORDS: Multigrid; massive Gaussian model; Monte Carlo; critical...
The Gauss Center Research in Multiscale Scientific Computation
 Elect. Trans. Numer. Anal
, 1997
"... . The recent research of the author and his collaborators on multiscale computational methods is reported, emphasizing main ideas and interrelations between various fields, and listing the relevant bibliography. The reported areas include: topefficiency multigrid methods in fluid dynamics; atmosph ..."
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Cited by 4 (1 self)
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. The recent research of the author and his collaborators on multiscale computational methods is reported, emphasizing main ideas and interrelations between various fields, and listing the relevant bibliography. The reported areas include: topefficiency multigrid methods in fluid dynamics; atmospheric data assimilation; PDE solvers on unbounded domains; wave/ray methods for highly indefinite equations; manyeigenfunction problems and abinitio quantum chemistry; fast evaluation of integral transforms on adaptive grids; multigrid Dirac solvers; fast inversematrix and determinant updates; multiscale MonteCarlo methods in statistical physics; molecular mechanics (including fast force summation, fast macromolecular energy minimization, MonteCarlo methods at equilibrium and the combination of smallscale equilibrium with largescale dynamics); image processing (edge detection and segmentation); and tomography. Key words. scientific computation, multiscale, multiresolution, multigrid,...
Methods of Systematic Upscaling
"... Systematic Upscaling (SU)is a new multiscale computational methodology for the accurate derivation of equations (or statistical relations) that govern a given physical system at increasingly larger scales. Starting at a fine (e.g., atomistic) scale where firstprinciple laws (e.g., differential equa ..."
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Cited by 4 (0 self)
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Systematic Upscaling (SU)is a new multiscale computational methodology for the accurate derivation of equations (or statistical relations) that govern a given physical system at increasingly larger scales. Starting at a fine (e.g., atomistic) scale where firstprinciple laws (e.g., differential equations) are known, SU advances, scale after scale, to obtain suitable variables and operational rules for simulating the system at any large scale of interest. SU combines the complementary advantages of two multilevel computational paradigms that have emerged over the last 35 years: multigrid in applied mathematics and renormalization group in theoretical physics. It includes systematic procedures to iterate back and forth between all the scales of the physical problem, with a general criterion for choosing appropriate variables that operate at each level, and general techniques to derive their governing relations. Indefinitely large systems can in this way be simulated, with computation at each level being needed only within limited representative windows. No scale separation is assumed; unlike conventional adhoc multiscale modeling, SU is in principle quite generally applicable, free of slowdowns and bears fullycontrolled accuracy. Fields that can be greatly impacted by SU range from elementary particle physics and quantum chemistry to molecular and macromolecular dynamics, material science, nanotechnology, biotechnology, and others. Cutting across such diverse fields, the present article does not focus on any specific application but on the generic principles of upscaling various types of systems, such as: problems defined on grids on one hand, and moving particles on the other hand; from ensembles of singleatom molecules to macromolecules in solution; local interactions as well as longrange ones; dynamical systems, both deterministic and stochastic; equilibrium calculations, including special procedures for low temperatures; energy minimization, particularly for functionals afflicted with multiscale nested attraction basins; etc. A suite of related upscaling techniques connects all these cases into one unified body of study. 1 1
Statistically Optimal Multigrid Algorithms for the Anharmonic Crystal Model
, 1997
"... Two types of multigrid algorithms for the one dimensional anharmonic crystal model are presented. The first type applies linear interpolation operators and the second type applies nonlinear interpolation operators with approximate Hamiltonians on coarse grids. For both algorithms, the question of el ..."
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Cited by 3 (1 self)
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Two types of multigrid algorithms for the one dimensional anharmonic crystal model are presented. The first type applies linear interpolation operators and the second type applies nonlinear interpolation operators with approximate Hamiltonians on coarse grids. For both algorithms, the question of eliminating the "volume" complexity factor is examined, i.e., the feasibility of the algorithm to remove the need to produce many independent finegrid configurations for averaging out their statistical deviations, so that thermodynamic limits can be calculated to relative accuracy " r in just O(" \Gamma2 r ) computer operations, where " r is the error relative to the standard deviation of the observable. The main difficulty arising in the nonlinear anharmonic crystal model is the coupling between different scales. In this paper, it is shown by analysis and numerical tests that the multigrid algorithm with the linear interpolation operators can eliminate the volume factor only partially, i.e...
Adaptive smoothed aggregation in lattice qcd
 Lecture Notes
, 2005
"... Summary. The linear systems arising in lattice QCD pose significant challenges for traditional iterative solvers. For physically interesting values of the socalled quark mass, these systems are nearly singular, indicating the need for efficient preconditioners. However, multilevel preconditioners c ..."
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Cited by 2 (2 self)
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Summary. The linear systems arising in lattice QCD pose significant challenges for traditional iterative solvers. For physically interesting values of the socalled quark mass, these systems are nearly singular, indicating the need for efficient preconditioners. However, multilevel preconditioners cannot easily be constructed because the Dirac operator associated with these systems has multiple nearkernel components that need to be approximated. Moreover, these components are generally both oscillatory and not known a priori. This paper presents the application of adaptive smoothed aggregation, αSA [2], to this Dirac system. Heuristic arguments and numerical results are provided to demonstrate that this recently developed extension of the smoothed aggregation methodology can be used to overcome the challenges posed by the Dirac system. 1