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Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations

Stochastic Manifolds for Nonlinear SPDEs II

  • Book
  • © 2015

Overview

Authors:
  1. Mickaël D. Chekroun

    1. University of California, Los Angeles, USA

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  2. Honghu Liu

    1. University of California, Los Angeles, USA

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  3. Shouhong Wang

    1. Indiana University, Bloomington, USA

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Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvii
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  2. General Introduction

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 1-7

  3. Preliminaries

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 9-17

  4. A Brief Review of the Results on Approximation of Stochastic Invariant Manifolds

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 19-24

  5. Pullback Characterization of Approximating, and Parameterizing Manifolds

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 25-58

  6. Non-Markovian Stochastic Reduced Equations

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 59-71

  7. Application to a Stochastic Burgers-Type Equation: Numerical Results

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 73-84

  8. Non-Markovian Stochastic Reduced Equations on the Fly

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 85-112

  9. Back Matter

    Pages 113-129
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • University of California, Los Angeles, USA

    Mickaël D. Chekroun, Honghu Liu

  • Indiana University, Bloomington, USA

    Shouhong Wang

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