Non-gaussian Test Models for Prediction and State Estimation with Model Errors

作     者：Michal BRANICKI Nan CHEN Andrew J.MAJDA 

作者机构：Department of Mathematics and Center for Atmosphere Ocean ScienceCourant Institute of Mathematical SciencesNew York University 

出 版 物：《Chinese Annals of Mathematics,Series B》 (数学年刊（B辑英文版）)

年 卷 期：2013年第34卷第1期

页      面：29-64页

核心收录：

学科分类：080704[工学-流体机械及工程] 07[理学] 080103[工学-流体力学] 08[工学] 0807[工学-动力工程及工程热物理] 070102[理学-计算数学] 0701[理学-数学] 0801[工学-力学（可授工学、理学学位）] 

基　　金：Project supported by the Office of Naval Research (ONR) Grants (No. ONR DRI N00014-10-1-0554) the DOD-MURI award "Physics Constrained Stochastic-Statistical Models for Extended Range Environmental Prediction" 

主　　题：Prediction Model error Information theory Feynman-Kac framework Fokker planck Turbulent dynamical systems 

摘      要：Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiqui- tous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scien- tific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Ka~ formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems： the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be use- ful for many other applications and algorithms for the real time prediction and the state estimation.