Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations

Stochastic Manifolds for Nonlinear SPDEs II

Mickaël D. Chekroun, Honghu Liu, Shouhong Wang (Autoren)

Buch | Softcover

XVII, 129 Seiten

2015 | 2015
Springer International Publishing (Verlag)
978-3-319-12519-0 (ISBN)

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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 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.

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.

"The monograph is well written and contains novel and important results for researchers in the field of analytical or numerical random dynamical systems and SPDEs. The clarity of presentation as well as the detailed list of references, makes it also appealing to research students as well as to newcomers to the field." (Athanasios Yannacopoulos, zbMATH 1331.37009, 2016)

Erscheint lt. Verlag	14.1.2015
Reihe/Serie	SpringerBriefs in Mathematics
Zusatzinfo	XVII, 129 p. 12 illus., 11 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	237 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	37L65,37D10,37L25,35B42,37L10,60H15,35R60 • Non-Markovian Reduced Equations • Ordinary differential equations • Partial differential equations • Pullback Characterization • Stochastic Burgers-Type Equation • Stochastic Parameterizing Manifolds, • Weak Non-Resonnance Conditions
ISBN-10	3-319-12519-2 / 3319125192
ISBN-13	978-3-319-12519-0 / 9783319125190
Zustand	Neuware