Parameterizing Manifolds and Non-Markovian Reduced Equations Stochastic Manifolds for Nonlinear Spdes II |
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Author:
| Chekroun, Mickaël D. Liu, Honghu Wang, Shouhong |
Series title: | SpringerBriefs in Mathematics Ser. |
ISBN: | 978-3-319-12519-0 |
Publication Date: | Jan 2015 |
Publisher: | Springer
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Book Format: | Paperback |
List Price: | AUD $111.95 |
Book Description:
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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...
More DescriptionIn 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.