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.

对许多现实世界中复杂系统的研究 - 例如，天气和气候，经济学和生物学 - 涉及对未来状态的某种物理系统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经常取决于合适的数学模型，通常与测量数据结合使用。在这里，挑战性问题以不可避免的错误或不确定性的形式出现，并且对模型的基本理论属性的理解有限。由于存在大量自由度，因此在复杂的物理系统中，所有这些挑战都会严重放大。该项目旨在提高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，尤其是在流体动力应用中引起的。具体而言，将解决以下主题：从稀疏和嘈杂的观测值中恢复缺失的物理参数；随机强迫模型的长期行为；并研究了某些确定性流体动力学模型的有限时间奇异性形成。这项研究将与几项教育活动集成，以促进学生的学习和专业发展机会，并组织有关统计抽样的研讨会，并参与学术界和行业。该项目的研究组成部分细分为以下特定项目：1）贝叶斯逆PDE问题和马尔可夫链蒙特卡洛（MCMC）算法。该项目将扩展到一般状态空间上的MCMC算法的发展理论，包括开发新算法和严格的合并结果。这些将应用于各种流体动力学示例的背景下，贝叶斯逆PDE问题所描述的稀疏和嘈杂数据的无限二维物理量。 2）随机PDE的混合速率和相关数值近似的弱收敛性。 PI将显示与几种随机流体模型相关的马尔可夫半群的瓦斯坦收缩，这意味着指数混合速率以及相关不变度度量的唯一性。 PI还将考虑这些模型的合适数值离散化，并在时间上显示均匀的收敛性和渐近数值偏差估计值。 3）分析流体动力模型中局部自相似的奇异性场景。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，可能发生有限的局部自相似类型的爆炸。 PI将将广义的表面准地球化方程视为范式，并分析耗散性和非疾病案例。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准来评估的支持。