Reduced-Order Multiscale Models for Uncertainty Quantification, Data Assimilation and Control

The mathematical study of turbulent flows requires models that can account for a large range of spatiotemporal scales, from small scale eddies to large scale coherent structures, the nonlinear interactions that are responsible for the transfer of energy across those scales, and statistical tools to account for the uncertainty and limited number of measurements available. Examples can be found in the study of atmospheric and oceanic flows, controlled plasma fusion, and other engineering applications. Due to their nonlinear coupling across a wide range of spatiotemporal scales, a rigorous analysis of these systems often becomes intractable and direct numerical simulations are likely to be expensive and inaccurate. The focus of this project is to develop a mathematical framework to derive tractable reduced-order models that effectively capture the dynamics of complex turbulent systems and apply them to complex systems of practical interest. This unified mathematical framework is based on the systematic integration of approaches from data assimilation, uncertainty quantification, and optimal control. The project will also provide training and research opportunities for undergraduate and graduate students. This project will develop a general framework for the formulation of self-consistent reduced-order closure models for turbulent flows, with theoretical justifications and application to relevant fluid flow systems. The unified reduced-order model is achieved through a precise decomposition of the state of the system into low-order statistical moments that characterize the dominant, leading-order coherent structures, coupled with the stochastic fluctuations modes accounting for the higher-order non-Gaussian statistics quantifying the multiscale feedback. The reduced-order model will form the basis for new multiscale data assimilation strategies with partial and noisy data. In addition, the reduced-order model will be used to formulate new mean-field control models to drive the fluid system to a desired coherent state. The resulting methods will be applied to several concrete models for geophysical flows and plasma physics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.