“Coarse” stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples

Makeev, Alexei G.; Maroudas, Dimitrios; Kevrekidis, Ioannis G.

doi:10.1063/1.1476929

Cited by 120 publications

(197 citation statements)

References 24 publications

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“…The coarse Newton method as well as coarse stability and bifurcation analyses have been described elsewhere 23,24,25 . In the context of the rare events problem, the coarse Newton method can be used to obtain the location of the saddle point.…”

Section: A Coarse Newton Methodsmentioning

confidence: 99%

Coarse-grained kinetic computations for rare events: Application to micelle formation

Kopelevich¹,

Panagiotopoulos²,

Kevrekidis³

2005

The Journal of Chemical Physics

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108

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We discuss a coarse-grained approach to the computation of rare events in the context of grand canonical Monte Carlo (GCMC) simulations of self-assembly of surfactant molecules into micelles.The basic assumption is that the computational system dynamics can be decomposed into two parts -fast (noise) and slow (reaction coordinates) dynamics, so that the system can be described by an effective, coarse grained Fokker-Planck (FP) equation. While such an assumption may be valid in many circumstances, an explicit form of FP equation is not always available. In our computations we bypass the analytic derivation of such an effective FP equation. The effective free energy gradient and the state-dependent magnitude of the random noise, which are necessary to formulate the effective Fokker-Planck equation, are obtained from ensembles of short bursts of microscopic simulations with judiciously chosen initial conditions. The reaction coordinate in our micelle formation problem is taken to be the size of a cluster of surfactant molecules. We test the validity of the effective FP description in this system and reconstruct a coarse-grained free energy surface in good agreement with full-scale GCMC simulations. We also show that, for very small clusters, the cluster size seizes to be a good reaction coordinate for a one-dimensional effective description. We discuss possible ways to improve the current model and to take higher-dimensional coarse-grained dynamics into account.

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Section: A Coarse Newton Methodsmentioning

confidence: 99%

Coarse-grained kinetic computations for rare events: Application to micelle formation

Kopelevich¹,

Panagiotopoulos²,

Kevrekidis³

2005

The Journal of Chemical Physics

Self Cite

108

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“…Previous studies [47,48] have already addressed how one can use the "equation-free" framework to find steady states through the zeros of a timestepper (a discrete time map resulting from an integrator scheme) using a Newton -Raphson (NR) method. Here we demonstrate how the information contained in our local diffusion model can be used to carry out such a NR scheme to solve the fixed point problems arising in implicit integration using the timestepper associated with the implicit Euler projective integration (the results of this integration scheme applied to the modified MM system were already shown in the previous section).…”

Section: Equation-free Fixed-point Methods For Integratorsmentioning

confidence: 99%

“…The drift component of our illustrative model corresponds to a limiting ODE that has two stable fixed-points for the parameter values used [47]. We used an SDE as our data generating process (our inner simulator) so that we can accurately calculate the invariant distribution of the model.…”

Section: Numerical Methods For Finding Extrema Of Distributionsmentioning

confidence: 99%

“…One should contrast this equation-free Newton method with that in Ref. [47] where the Jacobians of the Newton scheme were determined using a finite difference approximation coupled with least squares estimation. With noisy data, determining the appropriate finite difference stencil when the noise is not uniform in statespace (as it is in all applications shown here) is quite nontrivial.…”

Section: þ: ð25þmentioning

confidence: 99%

“…This is done because, in some systems, one knows that the system dynamics converge to a deterministic limit. When the functional form of this deterministic limit is unknown a priori (this can occur in kinetic Monte Carlo simulations with lateral interactions [47,48] as opposed to perfectly mixed SSA models) we could apply the same ideas presented here to estimate function values "on-demand". For the "inner simulators" of the systems studied, it is known that when V !…”

Section: Local Diffusion Models and Likelihood Expansionsmentioning

confidence: 99%

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Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics

Calderon

2007

Molecular Simulation

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Extracting coarse-grained derivative information from fine scale, atomistic/stochastic simulations constitutes an important component of multiscale numerics for reacting systems. In this paper, we demonstrate the use of local parametric diffusion models; observing the output of short bursts of stochastic simulation, the drift and diffusion coefficients of such local models are obtained "on demand" by maximizing an approximation of the likelihood function. The parametric fit utilizes the likelihood expansion of Aït-Sahalia, giving closed-form expressions for the transition density of both scalar and multivariate processes. The fine scale simulations generating the data are the Gillespie stochastic simulation algorithm as well as stochastic differential equations (SDEs). The postulated local parametric diffusion models are SDEs with affine drift and diffusion functions; the information extracted from these (estimated) local parametric models is utilized in various "equation-free" computational tasks. We demonstrate how such local diffusion modeling can (1) contribute to linking stochastic simulation with traditional, continuum numerical methods such as fixed point and initial value problem solving algorithms as well as variational integration; (2) aid in extracting quantitative information about the invariant distribution associated with the underlying stochastic system; and (3) be used in computational model reduction motivated by singular perturbation theory.

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Equation‐free: The computer‐aided analysis of complex multiscale systems

2004

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T he best available descriptions of systems often come at a fine level (atomistic, stochastic, microscopic, agent based), whereas the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization, and control) are at a much coarser, macroscopic level. Traditional modeling approaches start by deriving macroscopic evolution equations from microscopic models, and then bringing an arsenal of computational tools to bear on these macroscopic descriptions. Over the last few years with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the "equation-free" approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of computational "experiments". In traditional numerical analysis, the main code "pings" a subroutine containing the model, and uses the returned information (time derivatives, etc.) to perform computer-assisted analysis. In our approach the same main code "pings" a subroutine that runs an ensemble of appropriately initialized computational experiments from which the same quantities are estimated. Traditional continuum numerical algorithms can, thus, be viewed as protocols for experimental design (where "experiment" means a computational experiment set up, and performed with a model at a different level of description). Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation -through matrix-free numerical analysis, and systems theory tools like estimation-bridge microscopic simulation with macroscopic modeling. If enough control authority exists to initialize laboratory experiments "at will" this computational enabling technology can lead to experimental protocols for the equation-free exploration of complex system dynamics. The Equation-Free ApproachA persistent feature of many complex systems is the emergence of macroscopic, coherent behavior from the interactions of microscopic agents such as molecules, cells, or individuals in a population. The implication is that macroscopic rules (a description of the system at a coarse-grained, high level) can somehow be deduced from microscopic ones (a description at a much finer level). For laminar Newtonian fluid mechanics, a successful coarse-grained description (the Navier-Stokes equations) was known on a phenomenological basis long before its approximate derivation from kinetic theory. Today, we must frequently study systems for which the physics can be modeled at a microscopic, fine scale; yet, it is practically impossible to derive a good macroscopic description from the microscopic rules. Hence, we look to the computer to explore the macroscopic behavior, based on the microscopic description.

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“Coarse” stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples

Cited by 120 publications

References 24 publications

Coarse-grained kinetic computations for rare events: Application to micelle formation

Coarse-grained kinetic computations for rare events: Application to micelle formation

Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics

Equation‐free: The computer‐aided analysis of complex multiscale systems

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