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

概率框架提供了湍流流动的基本描述，另一方面也是量化这些流动的建模和测量中的不确定性的宝贵基础。虽然人们早就认识到概率方法是必不可少的，但在概率方法的分析和应用中，许多基本问题仍然没有得到解决。此外，最近理论和计算的重大发展，受噪声干扰的非均匀数据的不断增长，以及在气候科学和地球物理中应用对精细化统计工具的持续需求，使概率流体力学领域充满活力。在这种背景下，该项目承担了随机分析、非线性偏微分方程组和流体动力学的交叉研究计划。将考虑一系列与湍流有关的数学问题，这些问题既涉及理论基础，也涉及测量不确定性的量化。研究程序围绕着无限维马尔可夫系统及其不变度量(即统计稳定状态)的设置，作为确定统计量的稳健可观测性的一种手段。在这里，来自概率论、计算统计学、遍历理论和泛函分析的前沿工具为解决具有挑战性的公开问题和发展新的理论基础提供了统一的基础：1)非线性随机偏微分方程组的遍历不变测度理论。2)不变测度对物理参数和数值参数的依赖性。3)贝叶斯计算设置，用于结合噪声测量来估计流体流动中的不确定程度。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.