CAREER: Analysis of uncertainty, long-time statistics and singularity formation in fluid flow models

Drexel University, Philadelphia PA

Investigators

Abstract

The study of many real-world complex systems - e.g., in weather, economics, and biology - involve the prediction of future states of a certain physical system or estimation of unknown parameters. Methods for addressing these prediction or estimation questions frequently rely on suitable mathematical models often combined with measurement data. Here challenging issues arise in the form of unavoidable errors or uncertainties in both model and measurements, as well as a limited understanding of the underlying theoretical properties of the model. All such challenges are severely amplified in complex physical systems due to the presence of a large number of degrees of freedom. This project aims to advance rigorous understanding of these problems and develop new techniques in the context of high-dimensional complex systems, particularly arising in fluid dynamics applications. Specifically, the following topics will be addressed: recovery of missing physical parameters from sparse and noisy observations; long-time behavior of stochastically forced models; and investigation of finite-time singularity formation of certain deterministic hydrodynamic models. The research will be integrated with several educational activities to promote learning and professional development opportunities for students and the organization of a workshop on statistical sampling, with participation from both academia and industry. The research component of this project is subdivided into the following specific projects: 1) Bayesian inverse PDE problems and Markov Chain Monte Carlo (MCMC) algorithms. This project will expand on a developing theory of MCMC algorithms on general state spaces, including the development of new algorithms and rigorous convergence results. These will be applied in the recovery of infinite-dimensional physical quantities from sparse and noisy data as described by a Bayesian inverse PDE problem, in the context of various fluid dynamics examples. 2) Mixing rates for stochastic PDEs and weak convergence of associated numerical approximations. The PI will show Wasserstein contraction for the Markovian semigroup associated to several stochastic fluid models, a result that implies exponential mixing rates as well as uniqueness of the associated invariant measure. The PI will also consider suitable numerical discretizations of these models and show uniform in time weak convergence and asymptotic numerical bias estimates. 3) Analysis of locally self-similar singularity scenarios in hydrodynamic models. As a means of investigating reliability of mathematical models, the PI will analyze the possible occurrence of finite-time blowup of locally self-similar type in the context of hydrodynamic models. The PI will consider the generalized surface quasi-geostrophic equation as a paradigm and analyze both dissipative and non-dissipative cases. 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.