### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 29-64 |

Number of pages | 36 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Chinese Annals of Mathematics. Series B*,

*34*(1), 29-64. https://doi.org/10.1007/s11401-012-0759-3

**Non-Gaussian Test Models for Prediction and State Estimation with Model Errors.** / Branicki, Michal; Chen, Nan; Majda, Andrew J.

Research output: Contribution to journal › Article

*Chinese Annals of Mathematics. Series B*, vol. 34, no. 1, pp. 29-64. https://doi.org/10.1007/s11401-012-0759-3

}

TY - JOUR

T1 - Non-Gaussian Test Models for Prediction and State Estimation with Model Errors

AU - Branicki, Michal

AU - Chen, Nan

AU - Majda, Andrew J.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Feynman-Kac framework

KW - Fokker planck

KW - Information theory

KW - Model error

KW - Prediction

KW - Turbulent dynamical systems

UR - http://www.scopus.com/inward/record.url?scp=84873174304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873174304&partnerID=8YFLogxK

U2 - 10.1007/s11401-012-0759-3

DO - 10.1007/s11401-012-0759-3

M3 - Article

VL - 34

SP - 29

EP - 64

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

SN - 0252-9599

IS - 1

ER -