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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.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
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 48 (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.
Projecting to a slow manifold: Singularly perturbed systems and legacy codes, Part 2 (working title), in preparation
"... Abstract. We consider dynamical systems possessing an attracting, invariant “slow manifold ” that can be parameterized by a few observable variables. We present a procedure that, given a process for integrating the system step by step and a set of values of the observables, finds the values of the r ..."
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Cited by 34 (16 self)
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Abstract. We consider dynamical systems possessing an attracting, invariant “slow manifold ” that can be parameterized by a few observable variables. We present a procedure that, given a process for integrating the system step by step and a set of values of the observables, finds the values of the remaining system variables such that the state is close to the slow manifold to some desired accuracy. It should be noted that this is not equivalent to “integrating down to the manifold ” since the latter process may significantly change the values of the observables. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it (because the system is not necessarily expressed in a singular perturbation form). We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mthorder approximation to a point on the slow manifold in a sense to be defined. We then show how this criterion can be applied approximately when the system is defined by a legacy code rather than directly through closed form equations. This procedure can be valuable when one wishes to start a simulation of the detailed model on the slow manifold with particular values of observable variables characterizing the slow manifold.
From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model
 Simul
, 2005
"... Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. He ..."
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Cited by 18 (10 self)
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Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. Here we focus on the flagellated bacterium E. coli, for which a great deal is known about signal detection, transduction, and celllevel swimming behavior. We review the biological background on individual and populationlevel processes and discuss the velocityjump approach used for describing populationlevel behavior based on individuallevel intracellular processes. In particular, we generalize the momentbased approach to macroscopic equations used earlier [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361–391] to higher dimensions and show how aspects of the signal transduction and response enter into the macroscopic equations. We also discuss computational issues surrounding the bacterial pattern formation problem and technical issues involved in the derivation of macroscopic equations.
Constraintdefined manifolds: a legacy code approach to lowdimensional computation
 J. Sci. Comp
"... If the dynamics of an evolutionary differential equation system possess a lowdimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximat ..."
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Cited by 16 (7 self)
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If the dynamics of an evolutionary differential equation system possess a lowdimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of “slow ” variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the ChafeeInfante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.
A computational strategy for multiscale systems with applications to Lorenz 96 model
, 2004
"... ..."
Decentralised Autonomic Computing: Analysing SelfOrganising Emergent Behaviour Using Advanced Numerical Methods
 In: Proceedings of IEEE International Conference on Autonomic Computing (ICAC’05
, 2005
"... When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent be ..."
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Cited by 8 (6 self)
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When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent behaviour to achieve the requirements. A wellfounded and practically feasible approach to study overall system behaviour is a prerequisite for successful deployment. On one hand, formal proofs of correct behaviour and even predictions of the exact systemwide behaviour are practically infeasible due to the complex, dynamic, and often nondeterministic nature of selforganising emergent systems. On the other hand, simple simulations give no convincing arguments for guaranteeing systemwide properties. We describe an alternative approach that allows to analyse and assess trends in systemwide behaviour, based on socalled equationfree macroscopic analysis. This technique yields more reliable results about the systemwide behaviour, compared to mere observation of simulation results, at an affordable computational cost. Numerical algorithms act at the systemwide level and steer the simulations. This allows to limit the amount of simulations considerably. We illustrate the approach by studying a particular systemwide property of a decentralised control system for Automated Guided Vehicles and we outline a road map towards a general methodology for studying decentralised autonomic computing systems.
Patch dynamics with buffers for homogenization problems
 J. Computational Physics
, 2006
"... An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such timedependent multiscale problems, an “equationfree ” framework has been proposed, of which patch dynamics is an essential component. Patch dyna ..."
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Cited by 6 (2 self)
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An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such timedependent multiscale problems, an “equationfree ” framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the spacetime domain. To reduce the effect of the artificially introduced box boundaries, we use buffer regions to “shield ” the boundary artefacts from the interior of the domain for short time intervals. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity, and propose a simple heuristic to determine a sufficient buffer size. The algorithm performance is illustrated through a set of numerical examples, which include a nonlinear reactiondiffusion equation and the Kuramoto–Sivashinsky equation. 1 1