2019/01/17 by Nan Chen, Andrew J. Majda, Chen, Nan +3
Computer Science · Earth and Planetary Sciences · Environmental Science · Mathematics · #Climate variability and models #FOS: Mathematics #Meteorological Phenomena and Simulations #Nonlinear Dynamics and Pattern Formation #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1901.07318
34 pages, 9 figures
openalex publication_date 2019/01/17 · arxiv created 2019/01/26 · arxiv updated 2019/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Nonlinear dynamical stochastic models are ubiquitous in different areas. Excitable media models are typical examples with large state dimensions. Their statistical properties are often of great interest but are also very challenging to compute. In this article, a theoretical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous mathematical theories show the covariance decay behavior due to both local and nonlocal effects, which result from the diffusion and the mean field interaction, respectively. 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 surrogate 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.