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.
| Autorius: | Mickaël D. Chekroun, Shouhong Wang, Honghu Liu, |
| Serija: | SpringerBriefs in Mathematics |
| Leidėjas: | Springer Nature Switzerland |
| Išleidimo metai: | 2015 |
| Knygos puslapių skaičius: | 148 |
| ISBN-10: | 3319125192 |
| ISBN-13: | 9783319125190 |
| Formatas: | Knyga minkštu viršeliu |
| Kalba: | Anglų |
| Žanras: | Cybernetics and systems theory |
Parašykite atsiliepimą apie „Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II“
