LEAPS-MPS: Dynamical Parameter Estimation for Hydrodynamic Equations

Cuny Hunter College, New York NY

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). A fundamental feature in the modeling of physical phenomena is the presence of parameters that encode various relevant properties. In the study of fluid flows, parameters arise that capture the ability of the fluid to generate friction by shearing against itself or quantify how temperature differences within the fluid can drive its motion, such as when the sun heats the ocean and a flame heats a pot of water, or more generally represent external sources of energy, such as disturbances in the atmosphere due to large-scale disruptive volcanic eruptions and anthropogenic greenhouse gas effects. The role of these parameters is crucial in the understanding of turbulence, climate dynamics, and other engineering applications. Their precise determination is obtained empirically through the collection of data. Often the data collected is incomplete or corrupted by noise that may arise from the measurement procedure, or the devices used. This project systematically addresses the issue of the accurate estimation of parameters in the context of fluid flows, through the mathematical analysis and computational study of a data-driven approach geared to identify conditions under which recovery of the parameters can be provably guaranteed in several suitably idealized practical situations. The project also builds infrastructure for harnessing the data revolution through course redevelopment, provides training and mentoring opportunities for undergraduate and graduate students, and fosters a robust and diverse mathematical culture through a peer mentorship program. The project will carry out a mathematical study of a data-driven dynamical algorithm for recovering unknown parameters arising in partial differential equations of hydrodynamics via a paradigm of feedback control. In this approach, observations on the system are directly inserted into the prognostic equation as an exogenous term that enforces convergence towards the observations. Several practical scenarios of parameter estimation are conceived and systematically analyzed, which include the simultaneous recovery of multiple parameters and the use of various modes of observations. A key property underpinning the success of this strategy is the existence of a known nonlinear mechanism found in a large class of dissipative systems that allows small scale information to be asymptotically enslaved to large scale information. The application of this property to parameter estimation is novel. As a result, this study will shed light on the role of this mechanism for parameter estimation in contexts beyond fluid flow and provide a comprehensive exploration of the robustness and limitations of this approach. 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.