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Stochastic parameterizing manifolds and non-Markovian reduced equations : Stochastic manifolds for nonlinear SPDEs II / by Mickaël D. Chekroun, Honghu Liu, Shouhong Wang.
Springer Nature - Springer Mathematics and Statistics eBooks 2015 English International Available online
View online- Format:
- Book
- Author/Creator:
- Chekroun, Mickaël D., Author.
- Liu, Honghu, Author.
- Wang, Shouhong, Author.
- Series:
- SpringerBriefs in Mathematics, 2191-8198
- Language:
- English
- Subjects (All):
- Differential equations, Partial.
- Dynamics.
- Ergodic theory.
- Probabilities.
- Differential equations.
- Partial Differential Equations.
- Dynamical Systems and Ergodic Theory.
- Probability Theory and Stochastic Processes.
- Ordinary Differential Equations.
- Local Subjects:
- Partial Differential Equations.
- Dynamical Systems and Ergodic Theory.
- Probability Theory and Stochastic Processes.
- Ordinary Differential Equations.
- Physical Description:
- 1 online resource (141 p.)
- Edition:
- 1st ed. 2015.
- Place of Publication:
- Cham : Springer International Publishing : Imprint: Springer, 2015.
- Language Note:
- English
- Summary:
- 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.
- 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.
- Notes:
- Description based upon print version of record.
- Includes bibliographical references and index.
- ISBN:
- 3-319-12520-6
- OCLC:
- 908086301
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