|
|
| |
| Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media |
| |
Citation： |
Nan CHEN,Andrew J. MAJDA,Xin T. TONG.Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media[J].Chinese Annals of Mathematics B,2019,40(6):891~924 |
| Page view： 557
Net amount： 443 |
Authors： |
Nan CHEN; Andrew J. MAJDA;Xin T. TONG |
Foundation： |
This work was supported by the Office of Vice
Chancellor for Research and Graduate Education (VCRGE) at University
of Wisconsin-Madison, the Office of Naval Research Grant ONR MURI
N00014-16-1-2161, the Center for Prototype Climate Modeling (CPCM)
at New York University Abu Dhabi Research Institute and NUS Grant
R-146-000-226-133. |
|
|
| 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 \linebreak
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. |
Keywords： |
Large state dimensions, Diffusion, Mean field interaction, Spatialaveraging strategy, Efficiently sampling |
Classification： |
65C40, 68Q17, 60G20 |
|
|
Download PDF Full-Text
|
|
|
|
