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27
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 38 (16 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.
Resolve the multitude of microscale interactions to holistically discretise the stochastically forced Burgers’ partial differential equation
, 2008
"... ..."
Damping factors for the gaptooth scheme
 of Lecture Notes in Computational Science and Engineering
, 2004
"... An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such timedependent multiscale problems, the gaptooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in clos ..."
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Cited by 14 (4 self)
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An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such timedependent multiscale problems, the gaptooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain; it only uses appropriately initialized simulations of the available microscopic model in a number of small boxes. For some model problems, including numerical homogenization, the scheme is essentially equivalent to a finite difference scheme, provided we repeatedly impose appropriate algebraic constraints on the solution for each box. Here, we demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffers that “shield” over relatively short times the dynamics inside each box from boundary effects. We explore and quantify the behavior of these schemes systematically through the numerical computation of damping factors of the corresponding coarse timestepper, for which no closed formula is available. 1
Equationfree, multiscale computation for unsteady random diffusion. Multiscale Model
 Simul
, 2005
"... Abstract. We present an “equationfree ” multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A ..."
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Cited by 3 (1 self)
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Abstract. We present an “equationfree ” multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed finescale computation of such a problem requires discretization and solution of a large system of equations, and can be prohibitively timeconsuming. To circumvent this difficulty, we propose an equationfree approach, where the finescale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarsegrained variables (observables) is evaluated during the finescale computation, and “projective integration” is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of finescale solutions with “correct ” correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented. Key words. multiscale problem, diffusion in random media, stochastic modeling, equationfree.
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
unknown title
, 2009
"... Holistic discretisation ensures fidelity to dynamics in two spatial dimensions ..."
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Holistic discretisation ensures fidelity to dynamics in two spatial dimensions
CONTENTS
"... 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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M ULTIPHYSICS M ODELING PATCH DYNAMICS FOR MULTISCALE PROBLEMS
"... The engineering analysis and microscopic simulations required for predicting materials’ properties from atomistic descriptions require new approaches for predicting macroscopic properties. Patch dynamics bridges the gap between the time and space scales at which the microscopic models operate, helpi ..."
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The engineering analysis and microscopic simulations required for predicting materials’ properties from atomistic descriptions require new approaches for predicting macroscopic properties. Patch dynamics bridges the gap between the time and space scales at which the microscopic models operate, helping predict systemlevel behavior.