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

  • Michal Branicki
  • Nan Chen
  • Andrew J. Majda

Abstract

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous 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 scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac 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 useful for many other applications and algorithms for the real time prediction and the state estimation.

Keywords

Prediction Model error Information theory Feynman-Kac framework Fokker-Planck Turbulent dynamical systems 

Mathematics Subject Classification

60G25 60H10 60H30 82C31 94A15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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