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E-grāmata: Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II

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  • Formāts: PDF+DRM
  • Sērija : SpringerBriefs in Mathematics
  • Izdošanas datums: 23-Dec-2014
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319125206
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Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs IIPalielināt
  • Formāts: PDF+DRM
  • Sērija : SpringerBriefs in Mathematics
  • Izdošanas datums: 23-Dec-2014
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319125206
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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.

Recenzijas

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)

1 General Introduction
1(8)
2 Preliminaries
9(10)
2.1 Stochastic Evolution Equations
9(4)
2.2 Random Dynamical Systems
13(3)
2.3 Cohomologous Cocycles and Random Evolution Equations
16(3)
3 A Brief Review of the Results on Approximation of Stochastic Invariant Manifolds
19(6)
3.1 Approximation of Stochastic Critical Manifolds
19(4)
3.2 Approximation of Stochastic Hyperbolic Invariant Manifolds
23(2)
4 Pullback Characterization of Approximating, and Parameterizing Manifolds
25(34)
4.1 Pullback Characterization of Approximating Manifolds
26(8)
4.2 Stochastic Parameterizing Manifolds
34(7)
4.3 Parameterizing Manifolds as Pullback Limits
41(4)
4.4 Existence of Stochastic Parameterizing Manifolds via Backward-Forward Systems
45(8)
4.5 Existence of Parameterizing Manifolds in the Deterministic Case
53(6)
5 Non-Markovian Stochastic Reduced Equations
59(14)
5.1 Low-order Stochastic Reduction Procedure Based on Parameterizing Manifolds
61(4)
5.2 An Abstract Example of PM-Based Reduced System
65(2)
5.3 PM-Based Reduced Systems as Non-Markovian SDEs
67(6)
6 Application to a Stochastic Burgers-Type Equation: Numerical Results
73(12)
6.1 Parameterization Defect of hλ(1): Numerical Estimates
75(5)
6.2 Stochastic Reduced Equations Based on hλ(1)
80(1)
6.3 Modeling Performance Achieved by the Stochastic Reduced Equations Based on
81(4)
7 Non-Markovian Stochastic Reduced Equations on the Fly
85(28)
7.1 Reduced System on the Fly Based on hλ(2)
86(2)
7.2 Reduced System on the Fly for the Stochastic Burgers-Type Equation
88(8)
7.2.1 Reduced Equations on the Fly in Coordinate Form: Derivation
89(4)
7.2.2 Reduced Equations on the Fly in Coordinate Form: Numerical Integration
93(3)
7.3 Existence of as Pullback Limit, New Memory Terms, and New Non-resonance Conditions
96(9)
7.4 Parameterization Defect of hλ(q), for 2 ≤ q ≤ 10: Numerical Estimates
105(2)
7.5 Numerical Results: Reproduction of Probability Density and Autocorrelation Functions
107(6)
Appendix: Proof of Lemma 5.1 113(6)
References 119(8)
Index 127
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