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

许多现实世界的复杂系统的研究-例如， 在天气和气候，经济学和生物学中-涉及预测某个物理系统的未来状态或估计未知参数。解决这些预测或估计问题的方法通常依赖于通常与测量数据相结合的合适的数学模型。在这里，挑战性的问题出现在不可避免的错误或不确定性的形式在模型和测量，以及有限的理解模型的基本理论属性。由于存在大量的自由度，所有这些挑战在复杂的物理系统中被严重放大。该项目旨在推进对这些问题的严格理解，并在高维复杂系统的背景下开发新技术，特别是在流体动力学应用中。具体而言，将讨论以下主题：从稀疏和嘈杂的观测数据中恢复丢失的物理参数;随机强迫模型的长时间行为;以及某些确定性流体动力学模型的有限时间奇异性形成的调查。这项研究将与几项教育活动结合起来，以促进学生的学习和专业发展机会，并组织一个有学术界和工业界参加的统计抽样讲习班。本计画的研究内容分为以下几个具体的计画：1）贝氏逆偏微分方程问题与马尔可夫链蒙地卡罗演算法。这个项目将扩展一般状态空间上MCMC算法的发展理论，包括新算法的发展和严格的收敛结果。这些将被应用在恢复的无限维物理量从稀疏和嘈杂的数据所描述的贝叶斯逆PDE问题，在各种流体动力学的例子的背景下。2)随机偏微分方程的混合率和相关数值逼近的弱收敛。PI将显示Wasserstein收缩的马尔可夫半群与几个随机流体模型，一个结果，这意味着指数混合率以及相关的不变测度的唯一性。PI还将考虑这些模型的适当数值离散化，并显示时间上的一致弱收敛和渐近数值偏差估计。3)流体力学模型中局部自相似奇异性分析。作为研究数学模型可靠性的一种手段，PI将分析在流体动力学模型的背景下可能发生的局部自相似类型的有限时间爆破。PI将考虑广义表面准地转方程作为一个范例，并分析耗散和非耗散的情况下。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The study of many real-world complex systems - e.g., in weather and climate, 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.