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

Michal Branicki, Nan Chen, Andrew J. Majda

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)29-64
Number of pages36
JournalChinese Annals of Mathematics. Series B
Issue number1
StatePublished - Jan 2013


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

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


Dive into the research topics of 'Non-Gaussian Test Models for Prediction and State Estimation with Model Errors'. Together they form a unique fingerprint.

Cite this