‘Coarse’ integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods

Gear, C. W.; Kevrekidis, Ioannis G.; Theodoropoulos, Constantinos

doi:10.1016/s0098-1354(02)00020-0

Cited by 182 publications

(239 citation statements)

References 28 publications

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“…The similarities and differences are illustrated by the example in section 3. Once the evolution of r(τ ) has been obtained (by integrating (13)), we may reconstruct the full solution u(t) from (6). To do this, we must first determine g(τ ).…”

Section: Methods Of Slicesmentioning

confidence: 99%

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Reduction and reconstruction for self-similar dynamical systems

et al. 2003

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We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied.The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.Mathematics Subject Classification: 70G65, 76M60, 60G18, 34A26, 37J15, 65P99

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Section: Methods Of Slicesmentioning

confidence: 99%

“…Equation (14) is called the reconstruction equation, and may be integrated to find g(τ ). Then (9) may be integrated to give τ (t), and finally the solution u(t) is found from (6). Note that the only difference between this procedure and that given in [18] is that time has been rescaled according to dt = m(g)dτ .…”

Section: Methods Of Slicesmentioning

confidence: 99%

Reduction and reconstruction for self-similar dynamical systems

et al. 2003

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“…Whenever it is possible to define a few collective variables (CVs) that provide a coarse-grained description of the slow modes [1,2], it is also of great relevance to compute the associated free energy surface (FES). In order to draw such a surface a straightforward approach is often not possible due to high barriers or other sampling bottlenecks.…”

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confidence: 99%

Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method

2008

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“…(2) and given α i values], (iii) evolving the lifted, fine-scale initial conditions for certain time horizon, (iv) restricting the resulting fine-scale description to the coarse observables [finding the PC coefficients of the final state], and (v) repeating the procedure as necessary to perform specific scientific computation steps. This is a general approach that has been combined with various fine-scale models [19,25]; see Refs. [20,21].…”

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confidence: 99%

Coarse Graining the Dynamics of Coupled Oscillators

2006

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We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of non-identical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and oscillator natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics. [1,2,5,6,7,8], chemical [9,10], and physical systems [11,12]. However, even in this ideal limit, some basic questions including global, quantitative stability of asymptotic states, still remain open [13,14,15,16]. Many real-world systems consist of a large, finite number of non-identical entities, where statistical techniques for the continuum-limit are not directly applicable. Exploring and understanding the dynamics of such finite oscillator assemblies is an important topic (e.g., see Ref. [17]).We present a computer-assisted approach to modeling the coarse-grained dynamics of such large, finite oscillator assemblies at and near full synchronization. The premise is that there exist a small number of coarse-grained variables (observables) adequately describing the long-term dynamics, and that a closed evolution equation for these observables exists, but is not explicitly available. To account for oscillator variability within the assembly, we treat both the variable oscillator properties (here, natural frequencies ω) and the oscillator states (here, phase angles θ) as random variables. Recognizing a quick development of correlations between ω and θ, we express the latter as a polynomial expansion of the former (borrowing Wiener polynomial chaos (PC) tools [18]); the PC expansion coefficients are our coarse observables.Availability of the governing equations for the variables of interest is a prerequisite to modeling and computation. We circumvent this step using the recently-developed equation-free (EF) framework for complex, multiscale systems modeling [19,20,21]. In this framework we can perform system-level computational tasks without explicit knowledge of the coarse-grained equations; these unavailable equations are solved by designing, performing and processing the results of short bursts of appropriately initialized detailed (fine-scale, microscopic) simulations.We consider a paradigmatic model of coupled oscillators, the Kuramoto model, consisting of a population of N all-to-all, phase-coupled limit-cycle oscillators with i.i.d. ω with distribution function g(ω). This model has been extensively studied because of its simplicity and certain mathematical tractability, yet it is not merely a toy model. It appears as a normal form for general systems of coupled oscillators (e.g. Refs. [10,11]).We choose a Gaussian with standard deviation σ ω = 0.1 for g(ω); however, our approach is not limited to this particular choice, nor to the Kuramoto model. Due to rotational symmetry, the mean frequency Ω = i ω i /N can be...

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‘Coarse’ integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods

Cited by 182 publications

References 28 publications

Reduction and reconstruction for self-similar dynamical systems

Reduction and reconstruction for self-similar dynamical systems

Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method

Coarse Graining the Dynamics of Coupled Oscillators

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