From molecular dynamics to coarse self-similar solutions: a simple example using equation-free computation

Chen, L; Debenedetti, Pablo G.; Gear, C. W.; Kevrekidis, Ioannis G.

doi:10.1016/j.jnnfm.2003.12.007

Cited by 30 publications

(39 citation statements)

References 40 publications

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“…Such initial conditions are essential for the implementation of equation-free algorithms: algorithms that solve the reduced problem without ever deriving it in closed form [4,25,27]). Indeed, short bursts of appropriately initialized simulations can be used to perform long term prediction (projective and coarse projective integration) for the reduced problem, its stability and bifurcation analysis, as well as tasks like control and optimization.…”

Section: Discussionmentioning

confidence: 99%

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

Gear¹,

Kaper²,

Kevrekidis³

et al. 2005

SIAM J. Appl. Dyn. Syst.

131

178

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The long-term dynamics of many dynamical systems evolve on an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. Yet a simulation using the full model of the problem requires initial values for all variables. Given a set of values for the observables parameterizing the slow manifold, one needs a procedure for finding the additional values such that the state is close to the slow manifold to some desired accuracy. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it. We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a point on the slow manifold in sense to be defined. We then show how this criterion can be applied approximately when the system is defined by a legacy code rather than directly through closed form equations.

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Section: Discussionmentioning

confidence: 99%

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

Gear¹,

Kaper²,

Kevrekidis³

et al. 2005

SIAM J. Appl. Dyn. Syst.

131

178

View full text Add to dashboard Cite

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“…When macroscopic scale-invariant PDEs are explicitly available, template conditions can be used to derive dynamical equations (termed "MN-dynamics") for the rescaled selfsimilar solutions and similarity exponents 21 . The idea of employing template conditions can also be used to obtain renormalized self-similar macroscale solutions and similarity exponents for multiscale systems whose coarse-level PDEs are not explicitly known 19 . The number of template conditions depends on how many rescaling variables are needed to renormalize the physical solutions.…”

Section: Coarse Dynamic Renormalization (Cdr) For Multidimensional Ramentioning

confidence: 99%

“…The template condition has the following physical meaning: the rescaled marginal CDF ω U always has the same value m at the u-direction coordinate e for all time t. Applying this template to Eqn. (19) and assuming ∂ω ∂v (e, v, t) decays exponentially as v → ∞, we have…”

Section: Coarse Dynamic Renormalization (Cdr) For Multidimensional Ramentioning

confidence: 99%

“…The same approach can be used to evolve effective medium (homogenized) descriptions of reactiontransport problems 11,12 based on short bursts of finely For multiscale systems exhibiting scale invariance at the macroscopic level, renormalization techniques can be used to solve for self-similar solutions and their scalings 13 , see also 14,15 . Recently, dynamic renormalization (e.g., 16,17,18 ) was used in conjuction with equation-free computation to obtain coarse-grained selfsimilar solutions using short bursts of fine-level, direct simulation 19 . This coarse dynamic renormalization (CDR) method finds macroscopically self-similar solutions with the help of template functions 20,21,22,23 .…”

Section: Introductionmentioning

confidence: 99%

“…Studying the evolution of particle distributions using their ICDF as an observable is convenient for onedimensional problems in space, where suitable bases for representing one-dimensional monotonic curves over finite one-dimensional domains are readily available (see the examples in 3,4,19 ). The extension to corresponding observables in more than one dimension, however, is nontrivial: the CDF itself, not being a bijective mapping, does not have an inverse.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Equation-Free Particle-Based Computations: Coarse Projective Integration and Coarse Dynamic Renormalization in 2D

Zou

Kevrekidis

Ghanem

2006

Ind. Eng. Chem. Res.

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Equation-free approaches have been proposed in recent years for the computational study of multiscale phenomena in engineering problems where evolution equations for the coarse-grained, system-level behavior are not explicitly available. In this paper we study the dynamics of a diffusive particle system in a laminar shear flow, described by a two-dimensional Brownian motion; in particular, we perform coarse projective integration and demonstrate the particle-based computation of coarse self-similar and asymptotically self-similar solutions for this problem. We use marginal and conditional Inverse Cumulative Distribution Functions (ICDFs) as the macroscopic observables of the evolving particle distribution.

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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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From molecular dynamics to coarse self-similar solutions: a simple example using equation-free computation

Cited by 30 publications

References 40 publications

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

Equation-Free Particle-Based Computations: Coarse Projective Integration and Coarse Dynamic Renormalization in 2D

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

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