Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

Chekroun, Mickaël D.; Glatt-Holtz, Nathan

doi:10.1007/s00220-012-1515-y

Cited by 72 publications

(66 citation statements)

References 54 publications

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“…The present remarks demonstrate the underlying relationships between multilayer systems of ordinary differential equations (ODEs) and systems of integro-differential equations, and help one understand why the multilayer structure of an MSM is essential in constructing a class of stochastic differential equations (SDEs) susceptible to approximate a GLE. These observations are actually rooted in older mathematical ideas from the study of models that involve distributed delays [46,47]; such models arise in theoretical population dynamics and in the modeling of materials with memory [48,49,50], as well as in climate dynamics [51,52].…”

Section: 3mentioning

confidence: 99%

Data-driven non-Markovian closure models

Kondrashov

Chekroun

Ghil

2015

Physica D: Nonlinear Phenomena

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121

178

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This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori-Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR-MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a very broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The approach is applied to conceptual nonlinear models borrowed from climate dynamics and population dynamics. In both cases, it is shown that the resulting closure models are able to capture the main statistical features of the dynamics, even in presence of weak time-scale separation.Comment: 47 pages, 7 figure

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Section: 3mentioning

confidence: 99%

Data-driven non-Markovian closure models

Kondrashov

Chekroun

Ghil

2015

Physica D: Nonlinear Phenomena

Self Cite

121

178

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“…As one gets further away from the first criticality, a larger number of Koornwinder polynomials is typically required to dispose of good GK approximations of, already, the uncontrolled dynamics; see [9]. The numerical burden of the synthesis of controls at a nearly optimal cost-by solving the HJB equations corresponding to the relevant GK systems-becomes then quickly prohibitive, especially for the case of locally distributed controls 10 . One avenue to work within reduced state spaces of further reduced dimension compared to what would be required by a GK approximation, is to search for high-mode parameterizations that help reduce the residual energy contained in the unresolved modes, i.e.…”

Section: Discussionmentioning

confidence: 99%

“…The inclusion of time-lag terms are aimed to account for delayed responses of the modeled systems to either internal or external factors. Examples of such factors include incubation period of infectious diseases [41], wave propagation [10,56], or time lags arising in engineering [38] to name a few.…”

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confidence: 99%

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Chekroun¹,

Kröner²,

Liu³

2018

Hamilton-Jacobi-Bellman Equations

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Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation. 26 References 26

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“…For instance, Łukaszewicz, Real and Robinson [34] used the notion of Generalized Banach limit to construct the invariant measures for general continuous dynamical systems on metric spaces. Later, Chekroun and Glatt‐Holtz [7] improved the results of [39] and [34] to construct invariant measures for a broad class of dissipative autonomous dynamical systems. Recently, Łukaszewicz and Robinson [35] extended the result of [7] to construct invariant measures for dissipative non‐autonomous dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows

Zhao

Sang

2020

Z Angew Math Mech

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In this article, we first prove the existence of trajectory attractor for the threedimensional incompressible micropolar fluids. This trajectory attractor is the minimal compact trajectory attracting set for the natural translation semigroup on the trajectory space. Then we construct the trajectory statistical solution for this micropolar fluids via the natural translation semigroup and trajectory attractor. In our construction the trajectory statistical solution is a space-time probability measure, which is carried by the trajectory attractor and is invariant under the acting of the translation semigroup. K E Y W O R D S invariant measure, micropolar fluids, trajectory attractor, trajectory statistical solution M S C 2 0 1 0 35B41, 34D35, 76F20

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Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

Cited by 72 publications

References 54 publications

Data-driven non-Markovian closure models

Data-driven non-Markovian closure models

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

Using trajectory attractor to construct trajectory statistical solution for the 3D incompressible micropolar flows

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