Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations

Stochastic Manifolds for Nonlinear SPDEs II

von Mickaël D. Chekroun, Honghu Liu, Shouhong Wang

129 Seiten, Taschenbuch

Kurzbeschreibung des Verlags

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.

• Reihe: SpringerBriefs in Mathematics
• Themen: Mathematik und Naturwissenschaften Mathematik Mathematische Analysis, allgemein Differentialrechnung und -gleichungen
• ISBN: 9783319125190
• Sprache: Englisch
• Erscheinungsdatum: 14.01.2015
• Größe: 23.5 x 15.5 cm
• Verlag: Springer International Publishing