### 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 language | English (US) |
---|---|

Pages (from-to) | 891-924 |

Number of pages | 34 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 40 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2019 |

### Fingerprint

### 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

*Chinese Annals of Mathematics. Series B*,

*40*(6), 891-924. https://doi.org/10.1007/s11401-019-0166-0

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

Research output: Contribution to journal › Article

*Chinese Annals of Mathematics. Series B*, vol. 40, no. 6, pp. 891-924. https://doi.org/10.1007/s11401-019-0166-0

}

TY - JOUR

T1 - Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

AU - Chen, Nan

AU - Majda, Andrew J.

AU - Tong, Xin T.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

KW - 60G20

KW - 65C40

KW - 68Q17

KW - Diffusion

KW - Efficiently sampling

KW - Large state dimensions

KW - Mean field interaction

KW - Spatial averaging strategy

UR - http://www.scopus.com/inward/record.url?scp=85075164743&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075164743&partnerID=8YFLogxK

U2 - 10.1007/s11401-019-0166-0

DO - 10.1007/s11401-019-0166-0

M3 - Article

AN - SCOPUS:85075164743

VL - 40

SP - 891

EP - 924

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

SN - 0252-9599

IS - 6

ER -