04293nam 22007695 450 991029978180332120251116133853.03-319-12520-610.1007/978-3-319-12520-6(CKB)3710000000324518(EBL)1966859(OCoLC)908086301(SSID)ssj0001408402(PQKBManifestationID)11814577(PQKBTitleCode)TC0001408402(PQKBWorkID)11346474(PQKB)10516356(DE-He213)978-3-319-12520-6(MiAaPQ)EBC1966859(PPN)18315049X(EXLCZ)99371000000032451820141223d2015 u| 0engur|n|---|||||txtccrStochastic parameterizing manifolds and non-Markovian reduced equations Stochastic manifolds for nonlinear SPDEs II /by Mickaël D. Chekroun, Honghu Liu, Shouhong Wang1st ed. 2015.Cham :Springer International Publishing :Imprint: Springer,2015.1 online resource (141 p.)SpringerBriefs in Mathematics,2191-8198Description based upon print version of record.3-319-12519-2 Includes bibliographical references and index.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.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.SpringerBriefs in Mathematics,2191-8198Differential equations, PartialDynamicsErgodic theoryProbabilitiesDifferential equationsPartial Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12155Dynamical Systems and Ergodic Theoryhttps://scigraph.springernature.com/ontologies/product-market-codes/M1204XProbability Theory and Stochastic Processeshttps://scigraph.springernature.com/ontologies/product-market-codes/M27004Ordinary Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12147Differential equations, Partial.Dynamics.Ergodic theory.Probabilities.Differential equations.Partial Differential Equations.Dynamical Systems and Ergodic Theory.Probability Theory and Stochastic Processes.Ordinary Differential Equations.519.22Chekroun Mickaël D.authttp://id.loc.gov/vocabulary/relators/aut0Liu Honghuauthttp://id.loc.gov/vocabulary/relators/autWang Shouhongauthttp://id.loc.gov/vocabulary/relators/autMiAaPQMiAaPQMiAaPQBOOK9910299781803321Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations2512144UNINA