### Abstract

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction-diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker-Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

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

Pages (from-to) | 1-80 |

Number of pages | 80 |

Journal | Entropy |

Volume | 20 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2018 |

### Fingerprint

### Keywords

- Conditional Gaussian mixture
- Conditional Gaussian systems
- Conformation theory;model error
- Hybrid strategy
- Multiscale nonlinear stochastic systems
- Parameter estimation
- Physics-constrained nonlinear stochastic models
- Stochastically coupled reaction-diffusion models
- Superparameterization

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Entropy*,

*20*(7), 1-80. https://doi.org/10.3390/e20070509

**Conditional Gaussian systems for multiscale nonlinear stochastic systems : Prediction, state estimation and uncertainty quantification.** / Chen, Nan; Majda, Andrew.

Research output: Contribution to journal › Article

*Entropy*, vol. 20, no. 7, pp. 1-80. https://doi.org/10.3390/e20070509

}

TY - JOUR

T1 - Conditional Gaussian systems for multiscale nonlinear stochastic systems

T2 - Prediction, state estimation and uncertainty quantification

AU - Chen, Nan

AU - Majda, Andrew

PY - 2018/7/1

Y1 - 2018/7/1

N2 - A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction-diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker-Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

AB - A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction-diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker-Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

KW - Conditional Gaussian mixture

KW - Conditional Gaussian systems

KW - Conformation theory;model error

KW - Hybrid strategy

KW - Multiscale nonlinear stochastic systems

KW - Parameter estimation

KW - Physics-constrained nonlinear stochastic models

KW - Stochastically coupled reaction-diffusion models

KW - Superparameterization

UR - http://www.scopus.com/inward/record.url?scp=85050302934&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050302934&partnerID=8YFLogxK

U2 - 10.3390/e20070509

DO - 10.3390/e20070509

M3 - Article

VL - 20

SP - 1

EP - 80

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 7

ER -