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

概率框架提供了湍流流体流动的基本描述，同时在另一方面作为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，仍有大量的基本问题尚未解决。此外，概率流体动力学领域已活跃的重大最近的理论和计算的发展，不断增长的洪流的异质数据破坏的噪音和不断需要完善的统计工具的应用，在气候科学和地球物理学。在这种背景下，该项目进行了一项研究计划的交叉随机分析，非线性偏微分方程和流体动力学。一个不同的收集有关湍流的数学问题，解决这两个理论基础和量化的测量不确定性将被考虑。该研究计划围绕无限维马尔可夫系统及其不变测度（即统计稳定状态）的设置，作为确定统计量的鲁棒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尖端工具为解决具有挑战性的开放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的不变测度遍历理论。2)不变测度对物理参数和数值参数的依赖性。3)该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

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.