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，丰富的应用程序的建模和严格的数学分析的挑战。这项研究将为模拟化学混合、生物活性流体和地球物理系统提供重要的新方法。改进这些领域设置的方法将提供更有效的工具，以解决病原体或污染物传播等重要问题，或量化气候危害的程度和类型。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的不确定度。该项目涉及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。在计算基础设施方面的最新进展，结合新的数学公式和新发现的算法，使得贝叶斯方法能够扩展到新的物理约束反问题类别。通常解的计算能力是基于偏微分方程(PDE)的非线性前向映射的单个解的计算能力的数倍，其中估计涉及函数而不是有限的数值集合，即我们感兴趣的是估计无限维的未知参数。研究人员将在随机和泛函分析、高性能计算、非线性偏微分方程组和流体动力学的交叉点上开展研究计划。具体地说，研究人员将(1)考虑一系列具有无限维未知数的物理激励的偏微分方程组反问题；(2)开发适合于从无限维测量中高效采样的新算法；(3)发展某些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量的收敛速度；以及(4)分析无限维模型的大数据观测极限的一致性。该项目为从稀疏、不规则的数据中测量湍流流动提供了有效的框架，同时开发了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。

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.