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59
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 277 (32 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.
Extracting macroscopic dynamics: model problems and algorithms
 NONLINEARITY
, 2004
"... In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic ..."
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Cited by 111 (8 self)
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In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lowerdimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective lowdimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of timescales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVDbased methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptoticsbased mode elimination, coarse timestepping methods and transferoperator based methodologies.
The heterogeneous multiscale method: A review
 COMMUN. COMPUT. PHYS
, 2007
"... This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several applic ..."
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Cited by 104 (7 self)
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This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, microfluidics, solids, interface problems, stochastic problems, and statistically selfsimilar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The paper ends with a discussion on some of
Multilevel Generalized Forcedirected Method for Circuit Placement
 In Proc. Int’l Symp. on Phys. Design
, 2005
"... recently given the rapid increase of circuit complexity, increase of interconnect delay, and potential suboptimality of existing placement algorithms [13]. In this paper we present a generalized forcedirected algorithm embedded in mPL2's [12] multilevel framework. Our new algorithm, named mPL ..."
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Cited by 74 (17 self)
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recently given the rapid increase of circuit complexity, increase of interconnect delay, and potential suboptimality of existing placement algorithms [13]. In this paper we present a generalized forcedirected algorithm embedded in mPL2's [12] multilevel framework. Our new algorithm, named mPL5, produces the shortest wirelength among all published placers with very competitive runtime on the IBM circuits used in [29]. The new contributions and enhancements are: (1) We develop a new analytical placement algorithm using a density constrained minimization formulation which can be viewed as a generalization of the forcedirected method in [16]; (2) We analyze and identify the advantages of our new algorithm over the forcedirected method; (3) We successfully incorporate the generalized forcedirected algorithm into a multilevel framework which significantly improves wirelength and speed. Compared to Capo9.0, our algorithm mPL5 produces 8% shorter wirelength and is 2X faster. Compared to Dragon3.01, mPL5 has 3% shorter wirelength and is 12X faster. Compared to Fengshui5.0, it has 5% shorter wirelength and is 2X faster. Compared to the ultrafast placement algorithm: FastPlace, mPL5 produces 8% shorter wirelength but is 6X slower. A fast mode of mPL5 (mPL5fast) can produce 1% shorter wirelength than FastPlace1. 0 and is only 2X slower. Moreover, mPL5fast has demonstrated better scalability than FastPlace1.0.
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 69 (9 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.
Segmentation and boundary detection using multiscale intensity measurements
 IN: CVPR. VOLUME I., HAWAII
, 2001
"... Image segmentation is difficult because objects may differ from their background by any of a variety of properties that can be observed in some, but often not all scales. A further complication is that coarse measurements, applied to the image for detecting these properties, often average over prope ..."
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Cited by 57 (6 self)
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Image segmentation is difficult because objects may differ from their background by any of a variety of properties that can be observed in some, but often not all scales. A further complication is that coarse measurements, applied to the image for detecting these properties, often average over properties of neighboring segments, making it difficult to separate the segments and to reliably detect their boundaries. Below we present a method for segmentation that generates and combines multiscale measurements of intensity contrast, texture differences, and boundary integrity. The method is based on our former algorithm SWA, which efficiently detects segments that optimize a normalizedcutlike measure by recursively coarsening a graph reflecting similarities between intensities of neighboring pixels. In this process aggregates of pixels of increasing size are gradually collected to form segments. We intervene in this process by computing properties of the aggregates and modifying the graph to reflect these coarse scale measurements. This allows us to detect regions that differ by fine as well as coarse properties, and to accurately locate their boundaries. Furthermore, by combining intensity differences with measures of boundary integrity across neighboring aggregates we can detect regions separated by weak, yet consistent edges.
Nearoptimal detection of geometric objects by fast multiscale methods
 IEEE TRANS. INFORM. THEORY
, 2005
"... We construct detectors for “geometric” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detection thre ..."
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Cited by 41 (9 self)
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We construct detectors for “geometric” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detection threshold—i.e., the signal strength below which no method of detection can be successful for large dataset size. ii) The optimal computational complexity of a nearoptimal detector, i.e., the complexity required to detect signals slightly exceeding the detection threshold. We describe a general approach to such problems which covers several classes of geometrically defined signals; for example, with onedimensional data, signals having elevated mean on an interval, and, indimensional data, signals with elevated mean on a rectangle, a ball, or an ellipsoid. In all these problems, we show that a naive or straightforward approach leads to detector thresholds and algorithms which are asymptotically far away from optimal. At the same time, a multiscale geometric analysis of these classes of objects allows us to derive asymptotically optimal detection thresholds and fast algorithms for nearoptimal detectors.
An enhanced multilevel algorithm for circuit placement
 In Proc. of ICCAD
, 2003
"... This paper presents several important enhancements to the recently published multilevel placement package mPL [12]. The improvements include (i) unconstrained quadratic relaxation on small, noncontiguous subproblems at every level of the hierarchy; (ii) improved interpolation (declustering) based on ..."
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Cited by 30 (10 self)
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This paper presents several important enhancements to the recently published multilevel placement package mPL [12]. The improvements include (i) unconstrained quadratic relaxation on small, noncontiguous subproblems at every level of the hierarchy; (ii) improved interpolation (declustering) based on techniques from algebraic multigrid (AMG), and (iii) iterated Vcycles with additional geometric information for aggregation in subsequent Vcycles. The enhanced version of mPL, named mPL2, improves the total wirelength result by about 12 % compared to the original version. The attractive scalability properties of the mPL run time have been largely retained, and the overall run time remains very competitive. Compared to gordianldomino [25] on uniformcellsize IBM/ISPD98 benchmarks, a speedup of well over 8 × on large circuits ( ≥ 100, 000 cells or nets) is obtained along with an average improvement in total wirelength of about 2%. Compared to Dragon [32] on the same benchmarks, a speedup of about 5 × is obtained at the cost of about 4 % increased wirelength. On the recently published PEKO synthetic benchmarks, mPL2 generates surprisingly highquality placements — roughly 60 % closer to the optimal than those produced by Capo 8.5 and Dragon — in run time about twice as long as Capo’s and about 1/10th of Dragon’s.
Graph Minimum Linear Arrangement by Multilevel Weighted Edge Contractions
, 2006
"... The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a lineartime algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. ..."
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Cited by 21 (7 self)
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The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a lineartime algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short running time of the algorithm enabled experiments with very large graphs.