Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds For Nonlinear Spdes Ii: Stochastic Manifolds For Nonline by Mickaël D. ChekrounStochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds For Nonlinear Spdes Ii: Stochastic Manifolds For Nonline by Mickaël D. Chekroun

Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds For…

byMickaël D. Chekroun, Honghu Liu, Shouhong Wang

Paperback | January 14, 2015

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In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
Title:Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds For…Format:PaperbackDimensions:129 pagesPublished:January 14, 2015Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3319125192

ISBN - 13:9783319125190

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Table of Contents

General Introduction.- Preliminaries.- Invariant Manifolds.- Pullback Characterization of Approximating, and Parameterizing Manifolds.- Non-Markovian Stochastic Reduced Equations.- On-Markovian Stochastic Reduced Equations on the Fly.- Proof of Lemma 5.1.-References.- Index.

Editorial Reviews

"The monograph is well written and contains novel and important results for researchers in the field of analytical or numerical random dynamical systems and SPDEs. The clarity of presentation as well as the detailed list of references, makes it also appealing to research students as well as to newcomers to the field." (Athanasios Yannacopoulos, zbMATH 1331.37009, 2016)