CAREER: Ensemble Kalman Methods and Bayesian Optimization in Inverse Problems and Data Assimilation

将复杂的预测模型与数据混合在许多应用中是必不可少的，包括数值天气预报，气候科学，石油工程，信号处理和医学成像。反问题和数据同化中的正演模型和动力系统日益复杂所带来的挑战，可以通过开发无需导数且只需很少模型评估的计算方法来缓解。该项目涉及两个重要的家庭的成本效益，导数免费的算法：集合卡尔曼方法和贝叶斯优化。将建立严格的数学分析，这将有助于理解这些算法，确定其在高维反问题和数据同化的潜力和局限性。方法论的贡献将集中在新算法和计算框架的设计，以合并无导数优化与机器学习。这些新算法的数值实现将公开提供。除了逆问题和数据同化，首席研究员还将研究集合卡尔曼方法和贝叶斯优化在大规模科学计算问题中的潜力，其中梯度不可用或计算昂贵，以及在隐私受到关注的数据科学应用中。该项目的一个核心组成部分是教育和研究的一体化。首席研究员将参与新的导师计划，这是一项合作计划，旨在芝加哥为少数民族服务的社区学院建立数据科学课程。该计划将为来自代表性不足群体的社区大学学生转学到芝加哥大学创造途径。此外，调查人员将完成两本书，针对研究生和高年级本科生，将纳入本项目的主题。 指导研究生和开发研究生课程是该项目的核心部分，该项目将包括关于集合卡尔曼方法和贝叶斯优化的两个相互关联的研究方向。Ensemble Kalman方法是地球物理科学中的流行算法，其中它们通常与小的集合规模一起使用以保持模型评估的数量较低。该项目的第一个研究重点将开发一个新的综合非渐近分析的集合卡尔曼方法，严格解释何时以及为什么一个小的集合规模可能足够。以前的分析，而不是集中在大合奏渐近，不能解释这些算法的实际成功与一个小合奏大小。这一研究方向的方法学贡献将集中在推导融合集合卡尔曼方法和机器学习的原则框架，以及基于逆问题和数据同化的分层公式的新型正则化技术。 拟议的非渐近理论将建立这些新方法的合奏规模要求。该项目的第二个研究重点将通过开发新的几何感知内核，采集功能和收敛保证来推进图形和流形设置中的贝叶斯优化。PI将使用计算调和分析的工具来获得流形上随机过程的近似保证，并使用信息论的工具来获得错误指定模型下的遗憾界。最后，研究人员将探索协同的方式来结合联合收割机集合卡尔曼方法和贝叶斯优化，利用这两个家庭的算法的优势，以减轻其weakness.This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。

Blending complex predictive models with data is essential in many applications, including numerical weather forecasting, climate science, petroleum engineering, signal processing, and medical imaging. The challenges posed by the increasing complexity of forward models and dynamical systems in inverse problems and data assimilation can be mitigated by the development of computational methods that are derivative-free and require few model evaluations. This project is concerned with two important families of cost-efficient, derivative-free algorithms: ensemble Kalman methods and Bayesian optimization. Rigorous mathematical analyses will established which will contribute to the understanding of these algorithms, determining their potential and limitations in high-dimensional inverse problems and data assimilation. Methodological contributions will focus on the design of novel algorithms and computational frameworks to merge derivative-free optimization with machine learning. Numerical implementations of these new algorithms will be made publicly available. Beyond inverse problems and data assimilation, the principal investigator will also investigate the potential of ensemble Kalman methods and Bayesian optimization in large-scale scientific computing problems where gradients are unavailable or expensive to compute, and in data science applications where privacy is a concern. A central component of the project is the integration of education and research. The principal investigator will engage in the new Preceptor Program, a collaborative initiative to build data science curricula at minority-serving community colleges in Chicago. This program will create pathways for community college students from underrepresented groups to transfer to the University of Chicago. In addition, the investigator will complete two books aimed at graduate and upper-level undergraduate students that will incorporate topics drawn from this project. Mentorship of graduate students and development of graduate-level courses is a core part of the project.The project will consist of two interrelated research thrusts on ensemble Kalman methods and Bayesian optimization. Ensemble Kalman methods are popular algorithms in the geophysical sciences, where they are often used with a small ensemble size to keep the number of model evaluations low. The first research thrust of the project will develop a new comprehensive non-asymptotic analysis of ensemble Kalman methods that rigorously explains when and why a small ensemble size may suffice. Previous analyses have focused instead on large ensemble asymptotics that cannot explain the practical success of these algorithms with a small ensemble size. Methodological contributions of this research thrust will be focused on deriving principled frameworks to blend ensemble Kalman methods and machine learning, as well as novel regularization techniques based on hierarchical formulations of inverse problems and data assimilation. The proposed non-asymptotic theory will establish ensemble size requirements for these new methods. The second research thrust of the project will advance Bayesian optimization in graphical and manifold settings by developing new geometry-aware kernels, acquisition functions, and convergence guarantees. The PI will use tools from computational harmonic analysis to obtain approximation guarantees for stochastic processes on manifolds and tools from information theory to obtain regret bounds under mis-specified models. Finally, the investigator will explore synergistic ways to combine ensemble Kalman methods and Bayesian optimization, leveraging the strengths of both families of algorithms to mitigate their weaknesses.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.