Problem reduction, renormalization, and memory

Chorin, Alexandre J.; Stinis, Panos

doi:10.2140/camcos.2006.1.1

Cited by 102 publications

(129 citation statements)

References 43 publications

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“…While the Markovian term is usually rather straightforward to compute (see §4), the memory term computation is rather involved owing to the presence of the orthogonal dynamics evolution operator e tQL . In fact, it is the presence of this operator which makes, in general, the computation of MZ-reduced models prohibitively expensive (see Chorin & Stinis [4] for a thorough discussion).…”

Section: The Mori-zwanzig Formalismmentioning

confidence: 99%

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Stinis

2015

Proc. R. Soc. A.

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Model reduction for complex systems is a rather active area of research. For many real-world systems, constructing an accurate reduced model is prohibitively expensive. The main difficulty stems from the tremendous range of spatial and temporal scales present in the solution of such systems. This leads to the need to develop reduced models where, inevitably, the resolved variables do not exhibit (spatial and/or temporal) scale separation from the unresolved ones. We present a brief survey of recent results on the construction of Mori-Zwanzigreduced models for such systems. The construction is inspired by the concepts of scale dependence and renormalization which first appeared in the context of high-energy and statistical physics.

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Section: The Mori-zwanzig Formalismmentioning

confidence: 99%

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Stinis

2015

Proc. R. Soc. A.

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“…The first term in (5) is usually called Markovian since it depends only on the values of the variables at the current instant, the second is called "noise" and the third "memory". The meaning of the different terms appearing in (5) and a connection (and generalization) to the fluctuation-dissipation theorems of irreversible statistical mechanics can be found in [11]. If we write…”

Section: The Mori-zwanzig Formalismmentioning

confidence: 99%

Higher Order Mori–Zwanzig Models for the Euler Equations

Stinis¹

2007

Multiscale Model. Simul.

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In a recent paper [10], an infinitely long memory model (the tmodel) for the Euler equations was presented and analyzed. The model can be derived by keeping the zeroth order term in a Taylor expansion of the memory integrand in the Mori-Zwanzig formalism. We present here a collection of models for the Euler equations which are based also on the Mori-Zwanzig formalism. The models arise from a Taylor expansion of a different operator, the orthogonal dynamics evolution operator, which appears in the memory integrand. The zero, first and second order models are constructed and simulated numerically. The form of the nonlinearity in the Euler equations, the special properties of the projection operator used and the general properties of any projection operator can be exploited to facilitate the recursive calculation of even higher order models. We use our models to compute the rate of energy decay for the Taylor-Green vortex problem. The results are in good agreement with the theoretical estimates. The energy decay appears to be organized in "waves" of activity, i.e. alternating periods of fast and slow decay. Our results corroborate the assumption in [10], that the modeling of the 3D Euler equations by a few low wavenumber modes should include a long memory.

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“…An elegant framework for deriving the effective model is the Mori-Zwanzig projection [2][3][4], which has recently become an extremely important tool to simplify complex dynamical systems [5][6][7][8][9][10][11][12][13][14][15]. In particular, this derivation led to a set of generalized Langevin equation (GLEs), a typical result of the Mori-Zwanzig procedure.…”

Section: Introductionmentioning

confidence: 99%

Parametric reduced models for the nonlinear Schrödinger equation

Harlim

2015

Phys. Rev. E

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Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular,we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parameterization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

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Problem reduction, renormalization, and memory

Cited by 102 publications

References 43 publications

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Higher Order Mori–Zwanzig Models for the Euler Equations

Parametric reduced models for the nonlinear Schrödinger equation

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