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Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations

Stochastic Manifolds for Nonlinear SPDEs II

  • Mickaël D. Chekroun
  • Honghu Liu
  • Shouhong Wang

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 1-7
  3. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 9-17
  4. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 19-24
  5. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 25-58
  6. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 59-71
  7. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 73-84
  8. Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 85-112
  9. Back Matter
    Pages 113-129

About this book

Introduction

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.

Keywords

37L65,37D10,37L25,35B42,37L10,60H15,35R60 Non-Markovian Reduced Equations Pullback Characterization Stochastic Burgers-Type Equation Stochastic Parameterizing Manifolds, Weak Non-Resonnance Conditions

Authors and affiliations

  • Mickaël D. Chekroun
    • 1
  • Honghu Liu
    • 2
  • Shouhong Wang
    • 3
  1. 1.University of CaliforniaLos AngelesUSA
  2. 2.University of CaliforniaLos AngelesUSA
  3. 3.Indiana UniversityBloomingtonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-12520-6
  • Copyright Information The Author(s) 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-12519-0
  • Online ISBN 978-3-319-12520-6
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site