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

职业：流体流动模型中的不确定性、长期统计数据和奇点形成分析

• 批准号: 2239325
• 负责人: Cecilia Mondaini
• 金额: $ 48.14万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2028-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239325&HistoricalAwards=false
• 关键词: CAREER; Analysis; uncertainty; long; time

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.

对许多真实世界的复杂系统的研究--例如，在天气和气候、经济和生物方面--涉及对某一物理系统的未来状态的预测或未知参数的估计。解决这些预测或估计问题的方法通常依赖于适当的数学模型，通常与测量数据相结合。在这里，具有挑战性的问题以模型和测量中不可避免的误差或不确定性的形式出现，以及对模型的基本理论属性的有限理解。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都被严重放大。该项目旨在促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：从稀疏和噪声观测中恢复丢失的物理参数；随机强迫模型的长时间行为；以及研究某些确定性流体动力学模型的有限时间奇点形成。这项研究将与促进学生学习和职业发展机会的几项教育活动相结合，并将组织一次统计抽样讲习班，学术界和工业界都将参加。本课题的研究内容分为以下几个具体项目：1)贝叶斯PDE逆问题和马尔可夫链蒙特卡罗(MCMC)算法。这个项目将扩展一般状态空间上的MCMC算法的发展理论，包括新算法的发展和严格的收敛结果。这些将被应用于从稀疏和有噪声的数据中恢复无限维物理量，如在各种流体动力学例子的背景下的贝叶斯逆PDE问题所描述的。2)随机偏微分方程解的混合率及其相关数值逼近的弱收敛。PI将显示与几个随机流体模型相关的马尔可夫半群的Wasserstein压缩，这一结果意味着指数混合率以及相关不变测度的唯一性。PI还将考虑这些模型的适当的数值离散化，并在时间上显示一致的弱收敛和渐近的数值偏差估计。3)水动力模型中的局部自相似奇异性情景分析。作为研究数学模型可靠性的一种手段，PI将在水动力模型的背景下分析可能发生的局部自相似类型的有限时间井喷。PI将把广义地表准地转方程作为一个范例，分析耗散和非耗散两种情况。这一奖项反映了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。