Nonlinear Conditional Gaussian Systems for Complex Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification
Abstract
Multiscale nonlinear turbulent dynamical systems are ubiquitous in many areas. They are characterized by a large dimensional phase space and a large dimension of strong instabilities which transfer energy throughout the system. Many non-Gaussian characteristics, such as extreme events, intermittency and fat-tailed probability density functions (PDFs), are often observed in these systems. Key mathematical issues are their basic mathematical structural properties and qualitative features, short- and long-range forecasting, uncertainty quantification (UQ) and data assimilation. However, understanding and predicting these complex nonlinear dynamics systems are often extremely difficult, especially in large dimensions. Therefore, it is of practical importanceto develop an effective modeling framework that captures major nonlinear and non-Gaussian features of these complex systems while it remains computationally efficient and amenable to detailed mathematical analysis.In this project, nonlinear conditional Gaussian systems (NCGS) for understanding and predicting complex multiscale nonlinear stochastic systems are developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. Many complex nonlinear dynamical models belong to the NCGS, such as various large-scale dynamical models in turbulence and geophysics and quite a few stochastically coupled reaction-diffusion models in neuroscience and ecology. See a recent published paper in Entropy by the PI and the Co-PI for a gallery of examples. One desirable feature of the NCGS is that it allows closed analytical formulae for solving the conditional statistics andis thus computationally efficient. Recently, the PI and the Co-PI have made major breakthroughs by developing an extremely rapid algorithm for computing the strong non-Gaussian statistics of the NCGS with high accuracy in large dimensions. The work was published in PNAS. Notably, the number of views/downloads of this paper from the PNAS official website has been more than 12,300 times, which is a significant number for an applied math paper and indicates the general interest to the readers.The following topics are planned to be studied in this project to exploit this exciting new development:1). Extremely Rapid Data Assimilation and Parameter Estimation Algorithms with Accurate Prediction in Complex Systems.2). Data Assimilation, Prediction and Extreme Events in a new Simplified General Circulation Model (GCM) for the Coupled MJO and El Ni~o and Dynamical Walker Circulation.3). Using NCGS as a Fast Preconditioner for Even More Complex Systems.4). Applications to Large Complex Systems in Climate, Atmosphere and Ocean Science.5). Comparison of Using Machine Learning and NCGS in Predicting and Understanding Complex Turbulent Dynamical Systems.6). New Algorithms for Solving High-Dimensional Fokker-Plank Equations with Rich Applications.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 20, 2019
- Source ID
- N000141912680
Entities
People
- Andrew Majda
Organizations
- New York University
- Office of Naval Research
- United States Navy