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Stochastic parameterizing manifolds ...

Chekroun, Mickael D.

Stochastic parameterizing manifolds and non-markovian reduced equationsStochastic Manifolds for Nonlinear SPDEs II /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Stochastic parameterizing manifolds and non-markovian reduced equationsby Mickael D. Chekroun, Honghu Liu, Shouhong Wang.
Reminder of title:	Stochastic Manifolds for Nonlinear SPDEs II /
Author:	Chekroun, Mickael D.
other author:	Liu, Honghu.
Published:	Cham :Springer International Publishing :2015.
Description:	xvii, 129 p. :ill. (some col.), digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Stochastic partial differential equationsNumerical solutions.
Online resource:	http://dx.doi.org/10.1007/978-3-319-12520-6
ISBN:	9783319125206 (electronic bk.)

Stochastic parameterizing manifolds and non-markovian reduced equationsStochastic Manifolds for Nonlinear SPDEs II /
Chekroun, Mickael D.

Stochastic parameterizing manifolds and non-markovian reduced equationsStochastic Manifolds for Nonlinear SPDEs II /[electronic resource] :by Mickael D. Chekroun, Honghu Liu, Shouhong Wang. - Cham :Springer International Publishing :2015. - xvii, 129 p. :ill. (some col.), digital ;24 cm. - SpringerBriefs in mathematics,2191-8198. - SpringerBriefs in mathematics..

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.

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.

ISBN: 9783319125206 (electronic bk.)

Standard No.: 10.1007/978-3-319-12520-6doiSubjects--Topical Terms:

713101
Stochastic partial differential equations
--Numerical solutions.

LC Class. No.: QA274.25

Dewey Class. No.: 515.353

Stochastic parameterizing manifolds and non-markovian reduced equationsStochastic Manifolds for Nonlinear SPDEs II /
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245 1 0 $a Stochastic parameterizing manifolds and non-markovian reduced equations $h [electronic resource] : $b Stochastic Manifolds for Nonlinear SPDEs II / $c by Mickael D. Chekroun, Honghu Liu, Shouhong Wang.
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505 0 $a 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.
520 $a 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.
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