Stochastic Methods in Fluid Mechanics: Ergodic Properties, Statistical Sampling, and Uncertainty Quantification

Tulane University, New Orleans LA

Investigators

Abstract

Probabilistic frameworks provide fundamental descriptions of turbulent fluid flows while on the other hand serving as an invaluable basis for quantifying uncertainties in the modeling and measurement of these flows. While it has long been understood that a probabilistic approach is indispensable, a wealth of basic questions remain unresolved in the analysis and application of probabilistic methodologies. Moreover, the field of probabilistic fluid dynamics has been enlivened by significant recent theoretical and computational developments, the ever-growing torrent of heterogeneous data corrupted by noise and by the ongoing need for refined statistical tools for applications in climate science and in geophysics. Against this backdrop, this project undertakes a program of research at the intersection of stochastic analysis, nonlinear partial differential equations and fluid dynamics. A diverse collection of mathematical problems concerning turbulent flows addressing both theoretical foundations and the quantification of uncertainties in measurement will be considered. The research program centers around the setting of infinite dimensional Markovian systems and their invariant measures (i.e. statistically steady states) as a means of determining the robust observability of statistical quantities. Here cutting-edge tools from probability, computational statistics, ergodic theory and functional analysis provide a unified basis for tackling challenging open problems and developing novel theoretical foundations in: 1) The ergodic theory of invariant measures for nonlinear Stochastic PDEs. 2) The dependence of invariant measures on physical and numerical parameters. 3) The Bayesian computational setting for incorporating noisy measurements to estimate degrees of uncertainty in fluid flows. 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.