LEADER 04294nam 22007695 450 001 9910299781803321 005 20220404192032.0 010 $a3-319-12520-6 024 7 $a10.1007/978-3-319-12520-6 035 $a(CKB)3710000000324518 035 $a(EBL)1966859 035 $a(OCoLC)908086301 035 $a(SSID)ssj0001408402 035 $a(PQKBManifestationID)11814577 035 $a(PQKBTitleCode)TC0001408402 035 $a(PQKBWorkID)11346474 035 $a(PQKB)10516356 035 $a(DE-He213)978-3-319-12520-6 035 $a(MiAaPQ)EBC1966859 035 $a(PPN)18315049X 035 $a(EXLCZ)993710000000324518 100 $a20141223d2015 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aStochastic parameterizing manifolds and non-Markovian reduced equations $eStochastic manifolds for nonlinear SPDEs II /$fby Mickaël D. Chekroun, Honghu Liu, Shouhong Wang 205 $a1st ed. 2015. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2015. 215 $a1 online resource (141 p.) 225 1 $aSpringerBriefs in Mathematics,$x2191-8198 300 $aDescription based upon print version of record. 311 $a3-319-12519-2 320 $aIncludes bibliographical references and index. 327 $aGeneral 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. 330 $aIn this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation. 410 0$aSpringerBriefs in Mathematics,$x2191-8198 606 $aPartial differential equations 606 $aDynamics 606 $aErgodic theory 606 $aProbabilities 606 $aDifferential equations 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aDynamical Systems and Ergodic Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M1204X 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aOrdinary Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12147 615 0$aPartial differential equations. 615 0$aDynamics. 615 0$aErgodic theory. 615 0$aProbabilities. 615 0$aDifferential equations. 615 14$aPartial Differential Equations. 615 24$aDynamical Systems and Ergodic Theory. 615 24$aProbability Theory and Stochastic Processes. 615 24$aOrdinary Differential Equations. 676 $a519.22 700 $aChekroun$b Mickaël D$4aut$4http://id.loc.gov/vocabulary/relators/aut$0768294 702 $aLiu$b Honghu$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aWang$b Shouhong$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910299781803321 996 $aStochastic Parameterizing Manifolds and Non-Markovian Reduced Equations$92512144 997 $aUNINA