ROUGH-PATH FLUID DYNAMICS (RPFD)
Abstract
We propose the development of our new methodology for uncertainty quantification and reducing uncertainty in computational fluid dynamics. Recently, the PI, CO-PI, and their collaborators have formulated a variational theory of rough-path fluid dynamics (RPFD) as rough partial differential equations (RPDEs) on geometric rough paths (GRPs). Our variational principle applies the calculus of GRP to derive RPDEs for fluid dynamics on GRPs. Our RPFD theory transfers two defining properties of fluid dynamics, the Kelvin circulation principle and the Newtonian force law, into the realm of RPDEs. The RPFD theory includes stochastic advection by Lie transport (SALT) theory as a particular case. In recent work on the SALT modeling framework, the PI and co-authors have demonstrated that the combination of stochastic uncertainty quantification and data assimilation (i.e., the incorporation of observations) enhances accuracy by reducing uncertainty. In the SALT framework, proper orthogonal decomposition (POD) was used to coarse-grain and calibrate parameters. Furthermore, a particle filtering methodology was applied to assimilate noisy observations. Because the POD algorithm and particle filtering methodology are not immediately available for GRPs, we propose to develop new capabilities in our proposal. In particular, we plan to leverage new rapid developments in machine learning and GRPs to facilitate calibration and data assimilation in RPFD. Our modeling approach will be effective whenever a body of hydrodynamic transport data shows the characteristic signal of high power at low frequencies. This characteristic signal is often seen in flows in Nature, such as atmospheric and oceanic geophysical flows, and in three-dimensional fluid turbulence, in general. We expect that our rough-path methodology will benefit from available theoretical results such as stability, support theorems, large deviation principles, and splitting schemes, all of which are consequences of their characterization as random dynamical systems generated by the solutions of the RPDEs. While our RPFD will be a fruitful source of open problems for mathematics, our proposed project will also provide enhanced methods for data assimilation in fluid applications by using data structures defined on rough path properties, such as their signature and its moments.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 21, 2022
- Source ID
- FA86552117034XX0
Entities
People
- Darryl D. Holm
Organizations
- Air Force Office of Scientific Research
- Imperial College London
- United States Air Force