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56
Gene regulatory networks: A coarsegrained, equationfree approach to multiscale computation
 J. Chem. Phys
, 2006
"... Abstract: We present computerassisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equationfree analysis is the design and execution of appropriatelyinitialized short bursts of stochastic simulations; the results of these are processed to ..."
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Cited by 24 (10 self)
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Abstract: We present computerassisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equationfree analysis is the design and execution of appropriatelyinitialized short bursts of stochastic simulations; the results of these are processed to estimate coarsegrained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy Φ and of a statedependent effective diffusion coefficient D that characterize an unavailable effective FokkerPlanck equation. Additionally we illustrate the linking of equationfree techniques with continuation methods for performing a form of stochastic “bifurcation analysis”; estimation of mean switching times in the case of a bistable switch is also implemented in this equationfree context. The accuracy of our methods is tested by direct comparison with longtime stochastic simulations. This type of equationfree analysis appears to be a promising approach to computing features of the longtime, coarsegrained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations. 1
CoarseGrained Numerical Bifurcation Analysis of Lattice Boltzmann Models
, 2004
"... In this paper we study the coarsegrained bifurcation analysis approach proposed by I.G. Kevrekidis and collaborators in PNAS 97(18):9840–9843. We extend the results obtained in that paper for a onedimensional FitzhHughNagumo lattice Boltzmann model in several ways. First, we extend the coarsegra ..."
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Cited by 13 (12 self)
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In this paper we study the coarsegrained bifurcation analysis approach proposed by I.G. Kevrekidis and collaborators in PNAS 97(18):9840–9843. We extend the results obtained in that paper for a onedimensional FitzhHughNagumo lattice Boltzmann model in several ways. First, we extend the coarsegrained time stepper concept to enable the computation of periodic solutions and we use the more versatile NewtonPicard method rather than the Recursive Projection Method for the numerical bifurcation analysis. Second, we compare the obtained bifurcation diagram with the bifurcation diagrams of the corresponding macroscopic PDE and of the lattice Boltzmann model. Most importantly, we perform an extensive study of the influence of the lifting or reconstruction step on the minimal successful time step of the coarsegrained time stepper and the accuracy of the results. It is shown experimentally that this time step must often be much larger than the time it takes for the higherorder moments to become slaved by the lowestorder moment, which somewhat contradicts earlier claims.
Accuracy of Hybrid Lattice Boltzmann/Finite Difference Schemes for ReactionDiffusion Systems
, 2006
"... In this article we construct a hybrid model by spatially coupling a lattice Boltzmann model (LBM) to a finite difference discretization of the partial differential equation (PDE) for reactiondiffusion systems. Because the LBM has more variables (the particle distribution functions) than the PDE (on ..."
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Cited by 10 (2 self)
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In this article we construct a hybrid model by spatially coupling a lattice Boltzmann model (LBM) to a finite difference discretization of the partial differential equation (PDE) for reactiondiffusion systems. Because the LBM has more variables (the particle distribution functions) than the PDE (only the particle density), we have a onetomany mapping problem from the PDE to the LBM domain at the interface. We perform this mapping using either results from the ChapmanEnskog expansion or a pointwise iterative scheme that approximates these analytical relations numerically. Most importantly, we show that the global spatial discretization error of the hybrid model is one order less accurate than the local error made at the interface. We derive closed expressions for the spatial discretization error at steady state and verify them numerically for several examples on the onedimensional domain.
Article EntropyRelated Extremum Principles for Model Reduction of
, 2010
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Basic types of coarsegraining
 Model Reduction and Coarse–Graining Approaches for Multiscale Phenomena
, 2006
"... Summary. We consider two basic types of coarsegraining: the Ehrenfests ’ coarsegraining and its extension to a general principle of nonequilibrium thermodynamics, and the coarsegraining based on uncertainty of dynamical models and εmotions (orbits). Nontechnical discussion of basic notions and ..."
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Cited by 7 (3 self)
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Summary. We consider two basic types of coarsegraining: the Ehrenfests ’ coarsegraining and its extension to a general principle of nonequilibrium thermodynamics, and the coarsegraining based on uncertainty of dynamical models and εmotions (orbits). Nontechnical discussion of basic notions and main coarsegraining theorems are presented: the theorem about entropy overproduction for the Ehrenfests ’ coarsegraining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for εmotions of general dynamical systems including structurally unstable systems. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfests ’ coarsegraining. General theory of reversible regularization and filtering semigroups in kinetics is presented, both for linear and nonlinear filters. We obtain explicit expressions and entropic stability conditions for filtered equations. A brief discussion of coarsegraining by rounding and by small noise is also presented. 1
Slow invariant manifolds as curvature of the flow of dynamical systems
 International Journal of Bifurcation & Chaos
"... Considering trajectory curves, integral of ndimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean nspace it will be established in this article that the curvature of the flow, i.e., the curvature of the trajectory curves of any ndimensional dynamica ..."
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Cited by 6 (3 self)
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Considering trajectory curves, integral of ndimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean nspace it will be established in this article that the curvature of the flow, i.e., the curvature of the trajectory curves of any ndimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of highdimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the socalled Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua’s piecewise linear and cubic models of dimensions three, four, and five will be provided as tutorial examples exemplifying this method as well as those of highdimensional dynamical systems.
A variational principle for computing slow invariant manifolds in dissipative dynamical systems
 SIAM J. Sci. Comput
, 2011
"... Abstract. A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory o ..."
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Cited by 5 (3 self)
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Abstract. A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a twodimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higherdimensional chemical reaction mechanism are presented. Key words. Model reduction, slow invariant manifold, optimization, calculus of variations, extremum principle, curvature, chemical kinetics AMS subject classifications. 37N40, 37M99, 80A30, 92E20 1. Introduction. In
Dimension reduction method for ODE fluid models
 J. Comp. Phys
"... We develop a new dimension reduction method for large size ODE systems obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large size classical particle systems with negligibly small variations of partic ..."
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Cited by 4 (3 self)
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We develop a new dimension reduction method for large size ODE systems obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large size classical particle systems with negligibly small variations of particle concentration. We propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution. To illustrate the computational advantages of the proposed reduction method, we use it to solve a system of Smoothed Particle Hydrodynamic ODEs describing single phase and twophase layered Poiseuille flows driven by uniform and periodic (in space) body forces. For the single phase Poiseuille flow driven by the uniform force, the coarse solution was obtained with the zeroorder deconvolution. For the single phase flow driven by the periodic body force and for the twophase flows, the higherorder (the first and secondorder) deconvolutions were necessary to obtain a sufficiently accurate solution. Key words: model reduction, ODEs, multiscale modeling, coarse integration, upscaling, closure problem, deconvolution Preprint 1
An equationfree approach to coupled oscillator dynamics: the Kuramoto model example
, 2005
"... We present an equationfree multiscale approach to the computational study of the collective dynamics of the Kuramoto model [Chemical Oscillations, Waves, and Turbulence, SpringerVerlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of ..."
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Cited by 4 (1 self)
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We present an equationfree multiscale approach to the computational study of the collective dynamics of the Kuramoto model [Chemical Oscillations, Waves, and Turbulence, SpringerVerlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of coarsegrained “observables ” of the system: the first few moments of the oscillator phase angle distribution. We circumvent the derivation of explicit dynamical equations (approximately) governing the evolution of these coarsegrained macroscopic variables; instead we use the equationfree framework [Kevrekidis et al., Comm. Math. Sci. 1(4), 715 (2003)] to computationally solve these equations without obtaining them in closed form. In this approach, the numerical tasks for the conceptually existing but unavailable coarsegrained equations are implemented through short bursts of appropriately initialized simulations of the “finescale”, detailed coupled oscillator model. Coarse projective integration and coarse fixed point computations are illustrated. Coupled nonlinear oscillators can exhibit spontaneous emergence of order, a fundamental qualitative feature of many complex dynamical systems [Manrubia et al., 2004]. The collective,
Accuracy and Stability of the Coarse TimeStepper for a Lattice Boltzmann Model
, 2007
"... The equationfree framework for multiscale computing is built around the central idea of a coarse timestepper, which is an approximate time integrator for the unavailable macroscopic model when only a microscopic simulator is given. In this paper, we study the numerical properties of the coarse tim ..."
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Cited by 3 (3 self)
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The equationfree framework for multiscale computing is built around the central idea of a coarse timestepper, which is an approximate time integrator for the unavailable macroscopic model when only a microscopic simulator is given. In this paper, we study the numerical properties of the coarse timestepper when a lattice Boltzmann model for onedimensional diffusion is used as the microscopic simulator. We derive analytical expressions for the accuracy and stability of the coarse timestepper, which allow us to study the influence of various aspects involved its construction.