Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions

Makeev, Alexei G.; Maroudas, Dimitrios; Panagiotopoulos, Athanassios Z.; Kevrekidis, Ioannis G.

doi:10.1063/1.1512274

Cited by 91 publications

(104 citation statements)

References 20 publications

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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%

“…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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Section: Equation-free Fixed-point Methods For Integratorsmentioning

confidence: 99%

Section: Local Diffusion Models and Likelihood Expansionsmentioning

confidence: 99%

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

Calderon

2007

Molecular Simulation

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“…In this context macrosocopic, coarse-grained equations are not explicitly available; yet we believe they exist, and we do have available a fine-scale (in this case, stochastic) dynamic simulator. We can then substitute the (unavailable) deterministic timestepper with a fine scale, stochastic timestepper involving lifting, evolving, and restriction steps 3,6,27 .…”

Section: A Numerical Bifurcation Computationsmentioning

confidence: 99%

“…(4) - (6) are well approximated by a system of two reaction-diffusion PDEs for the species concentrations U and V ; at the mesh points…”

Section: Deterministic Analysis Of the Model Problemmentioning

confidence: 99%

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Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

Qiao

Erban

Kelley

et al. 2006

The Journal of Chemical Physics

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Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging, is often a challenging task; approximate PDEs are typically obtained through mathematical closure procedures (e.g. mean-field approximations). In this paper, we show how such approximate macroscopic PDEs can be exploited in constructing preconditioners to accelerate stochastic simulations for spatially distributed particle-based process models. We illustrate how such preconditioning can improve the convergence of equation-free coarse-grained methods based on coarse timesteppers. Our model problem is a stochastic reaction-diffusion model capable of exhibiting Turing instabilities.

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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 bifurcation analysis of kinetic Monte Carlo simulations: A lattice-gas model with lateral interactions

Cited by 91 publications

References 20 publications

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

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

Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

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

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