Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nan Chen, Andrew J. Majda, Xin T. Tong

Research output: Contribution to journalArticle

Abstract

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.

Original languageEnglish (US)
Pages (from-to)891-924
Number of pages34
JournalChinese Annals of Mathematics. Series B
Volume40
Issue number6
DOIs
StatePublished - Nov 1 2019

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Excitable Media
Dynamical Model
Stochastic models
Covariance matrix
Stochastic Model
Averaging
Linear Model
Sampling
Localized Structures
Surrogate Model
Data Assimilation
FitzHugh-Nagumo
Exponential Decay
Mathematical Analysis
Mean Field
Coupled System
Statistical property
Higher Dimensions
Large scale systems
Nonlinear systems

Keywords

  • 60G20
  • 65C40
  • 68Q17
  • Diffusion
  • Efficiently sampling
  • Large state dimensions
  • Mean field interaction
  • Spatial averaging strategy

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media. / Chen, Nan; Majda, Andrew J.; Tong, Xin T.

In: Chinese Annals of Mathematics. Series B, Vol. 40, No. 6, 01.11.2019, p. 891-924.

Research output: Contribution to journalArticle

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