Collaborative Research: Bayesian Inversion Approaches to Partial Differential Equations: Theory, Algorithm Development, and Applications

该项目将有助于一个新的和快速发展的应用数学领域丰富的应用建模和挑战严格的数学分析。这项研究将为化学混合、生物活性流体和地球物理系统的建模提供重要的新方法。改进这些领域设置的方法将为解决诸如病原体或污染物的传播或量化气候危害的程度和类型等重要问题提供更有效的工具。研究人员将开发新的方法来校准和设计这些不同流体系统的有效测量策略，同时解决这些测量中固有不确定度的程度。这个项目需要一些研究生和其他早期职业科学家的培训和积极参与。该项目的跨机构和跨学科性质将为参与者提供独特的机会。计算基础设施的最新进展与新颖的数学公式和新发现的算法相结合，使得贝叶斯方法可以扩展到新的物理约束逆问题类别。解通常需要的计算能力是基于偏微分方程（PDEs）的非线性正演映射的单次解的许多倍，其中估计涉及一个函数而不是数值的有限集合，即我们对估计无限维未知参数感兴趣。研究人员将在随机和功能分析、高性能计算、非线性偏微分方程和流体动力学的交叉领域进行研究。具体而言，研究人员将(1)考虑一系列具有无限维未知数的物理动机PDE逆问题；(2)开发适用于从无限维测量中有效采样的新算法；(3)发展了一类无限维马尔可夫链蒙特卡罗（MCMC）算法的遍历理论，以严格评估采样目标后验测度的收敛率；(4)分析无限维模型大数据观测极限的一致性。该项目为从稀疏、不规则数据中测量湍流流体流动提供了有效的框架，同时开发了在计算统计学和数据科学中更广泛感兴趣的采样方法。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

The project will contribute to a new and rapidly developing area of applied mathematics rich with applications for modeling and challenges for rigorous mathematical analysis. This research will yield important new methods for modeling chemical mixing, biologically active fluids, and geophysical systems. Improving methods for these domain settings will provide more effective tools to address important problems such as the spread of pathogens or pollutants or to quantify the degree and type of climate hazards. The investigators will develop new methodologies for calibrating and designing effective measurement strategies of these various fluid systems, which simultaneously resolve degrees of the inherent uncertainty in these measurements. This project involves the training and active participation of a number of graduate students and other earlier career scientists. The cross institutional and cross disciplinary nature of this project will provide unique opportunities for the participants. Recent advances in computational infrastructure combined with novel mathematical formulations and newly discovered algorithms have allowed the extension of the Bayesian approach to new classes of physics-constrained inverse problems. Solutions typically take many times the computational power of a single solve of a nonlinear forward map based on a partial differential equations (PDEs) where the estimation concerns a function rather than a finite collection of numerical values, namely where we are interested in estimating an infinite-dimensional unknown parameter. The investigators will undertake a research program at the intersection of stochastic and functional analysis, high-performance computing, nonlinear PDEs, and fluid dynamics. Specifically, the investigators will (1) consider a series of physically motivated PDE inverse problems with infinite-dimensional unknowns; (2) develop novel algorithms adapted to efficiently sample from infinite-dimensional measures; (3) develop the ergodic theory for certain classes of infinite-dimensional Markov Chain Monte Carlo (MCMC) algorithms to rigorously assess rates of convergence in sampling target posterior measures; and (4) analyze consistency in the large data observation limit for infinite dimensional models. The project contributes effective frameworks for the measurement of turbulent fluid flows from sparse, irregular data while developing sampling methods of broader interest across computational statistics and data science.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.