ENGLISH

Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II

By Shouhong Wang (auth.) , Mickaël D. Chekroun , Honghu Liu

Book information

Publisher
Springer International Publishing
Year
2015
ISBN
978-3-319-12519-0, 978-3-319-12520-6
DOI
10.1007/978-3-319-12520-6
Language
english
Format
PDF
Filesize
4 MB (4586963 bytes)
Series
SpringerBriefs in Mathematics
Edition
1
Pages
129\141
Time added
2015-02-17 02:00:00
Partial Differential Equations; Dynamical Systems

Description

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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