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23
Application of coarse integration to bacterial chemotaxis
 SIAM J Appl Math
, 2005
"... Abstract. We have developed and implemented a numerical evolution scheme for a class of stochastic problems in which the temporal evolution occurs on widely separated time scales and for which the slow evolution can be described in terms of a small number of moments of an underlying probability dist ..."
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Abstract. We have developed and implemented a numerical evolution scheme for a class of stochastic problems in which the temporal evolution occurs on widely separated time scales and for which the slow evolution can be described in terms of a small number of moments of an underlying probability distribution. We demonstrate this method via a numerical simulation of chemotaxis in a population of motile, independent bacteria swimming in a prescribed gradient of a chemoattractant. The microscopic stochastic model, which is simulated using a Monte Carlo method, uses a simplified deterministic model for excitation/adaptation in signal transduction, coupled with a realistic, stochastic description of the flagellar motor. We show that projective time integration of “coarse” variables can be carried out on time scales long compared to those of microscopic dynamics. Our coarse description is based on the spatial cell density distribution. Thus we are assuming that the system “closes ” on this variable so that it can be described on long time scales solely by the spatial cell density. Computationally, the variables are the components of the density distribution expressed in terms of a few basis functions, given by the singular vectors of the spatial density distribution obtained from a sample Monte Carlo time evolution of the system. We present numerical results and analysis of errors in support of the efficacy of this timeintegration scheme.
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 14 (4 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
Modulated Fourier expansions and heterogeneous multiscale methods,
 IMA J. Numer. Anal.
, 2009
"... Abstract We show that, for highlyoscillatory ordinary differential equations problems, the modulated Fourier expansion approach can be advantageously used to understand and analyze the Heterogenous Multiscale Methods introduced by E, Engquist and their coworkers. ..."
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Cited by 10 (3 self)
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Abstract We show that, for highlyoscillatory ordinary differential equations problems, the modulated Fourier expansion approach can be advantageously used to understand and analyze the Heterogenous Multiscale Methods introduced by E, Engquist and their coworkers.
Parameter Estimation for Rough Differential Equations
, 2008
"... We construct an estimator based on “signature matching ” for differential equations driven by rough paths and we prove its consistency and asymptotic normality. Note that the the Moment Matching estimator is a special case of this estimator. ..."
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Cited by 3 (1 self)
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We construct an estimator based on “signature matching ” for differential equations driven by rough paths and we prove its consistency and asymptotic normality. Note that the the Moment Matching estimator is a special case of this estimator.
Finite difference patch dynamics for advection homogenization problems
 Model Reduction and CoarseGraining Approaches for Multiscale Phenomena
, 2006
"... Summary. We consider problems in which there is a separation between the (microscopic) scale at which the available model is defined, and the (macroscopic) scale of interest. For timedependent multiscale problems of this type, an “equationfree” framework has been proposed, of which patch dynamic ..."
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Cited by 1 (1 self)
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Summary. We consider problems in which there is a separation between the (microscopic) scale at which the available model is defined, and the (macroscopic) scale of interest. For timedependent multiscale problems of this type, 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 only uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover a fraction of the spacetime domain. We review some recent convergence results and demonstrate that the method allows to simulate advectiondominated problems accurately. 1
SOME CRITICAL ISSUES FOR THE “EQUATIONFREE” APPROACH TO MULTISCALE MODELING
"... The “equationfree” approach has been proposed in recent years as a general framework for developing multiscale methods for efficiently capturing the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the “eq ..."
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The “equationfree” approach has been proposed in recent years as a general framework for developing multiscale methods for efficiently capturing the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the “equationfree” umbrella, the projective integrators and the patch dynamics. We discuss some very simple examples in the context of the “equationfree ” approach. These examples seem to indicate that while its general philosophy is quite attractive and indeed similar to many other approaches in concurrent multiscale modeling, there are severe limitations to the specific implementation proposed by this approach.
Implicit Methods for EquationFree Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models. arXiv.org
, 2013
"... Abstract. First, we give a rigorous convergence result for equationfree analysis in the setting of slowfast systems using implicit lifting. Second, we apply this result to study the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. It is ..."
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Abstract. First, we give a rigorous convergence result for equationfree analysis in the setting of slowfast systems using implicit lifting. Second, we apply this result to study the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. It is shown, that the implicitly defined coarselevel time stepper converges to the true dynamics on the slow manifold with an accuracy that is beyond all orders of the small parameter measuring time scale separation. These results are applied to investigate the behavior of the microscopic traffic model on a macroscopic level. The traffic jams are waves that travel slowly and in opposite direction compared to the car velocity. The standard deviation is chosen as a macroscopic measure of traveling wave solutions and is continued on the macroscopic level in the equationfree setup. The collapse of the traffic jam to the free flow corresponds in the relevant parameter region at the macroscopic level to a saddlenode bifurcation of the traveling wave. We continue this bifurcation point in two parameters using equationfree analysis.
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"... Any use of trade, product, or finn names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Government ..."
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Any use of trade, product, or finn names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Government
behavior from individualbased models
, 2006
"... equationfree computational approach for extracting populationlevel ..."
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