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

职业：反问题和数据同化中的集成卡尔曼方法和贝叶斯优化

基本信息

批准号：
2237628
负责人：
Daniel Sanz-Alonso
金额：
$ 45万
依托单位：
University of Chicago
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：
2023-02-01 至 2028-01-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237628&HistoricalAwards=false
关键词：
CAREER Ensemble Kalman Methods Bayesian

项目摘要

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.

将复杂的预测模型与数据相结合在许多应用中至关重要，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。逆问题和数据同化中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导数且需要很少模型评估的计算方法来缓解。该项目涉及两个重要的经济高效、无导数算法系列：集成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，确定它们在高维逆问题和数据同化中的潜力和局限性。方法论贡献将集中于新颖算法和计算框架的设计，以将无导数优化与机器学习相结合。这些新算法的数值实现将公开。除了逆问题和数据同化之外，首席研究员还将研究集成卡尔曼方法和贝叶斯优化在梯度不可用或计算成本昂贵的大规模科学计算问题中以及在隐私受到关注的数据科学应用中的潜力。该项目的核心组成部分是教育和研究的整合。首席研究员将参与新的 Preceptor 计划，这是一项在芝加哥为少数族裔服务的社区学院建立数据科学课程的合作计划。该计划将为来自代表性不足群体的社区学院学生转学到芝加哥大学创造途径。此外，研究人员还将完成两本针对研究生和高年级本科生的书，其中将纳入来自该项目的主题。研究生指导和研究生水平课程的开发是该项目的核心部分。该项目将包括关于集成卡尔曼方法和贝叶斯优化的两个相互关联的研究重点。集成卡尔曼方法是地球物理科学中流行的算法，它们通常与较小的集成尺寸一起使用，以保持较低的模型评估数量。该项目的第一个研究主旨将开发一种新的综合卡尔曼方法的综合非渐近分析，该方法严格解释何时以及为何小规模的集合可能就足够了。之前的分析主要集中在大型系综渐近上，无法解释这些算法在小系综规模下的实际成功。这项研究的方法论贡献将集中于推导融合集成卡尔曼方法和机器学习的原则框架，以及基于反问题和数据同化的分层公式的新颖正则化技术。所提出的非渐近理论将为这些新方法建立集合大小要求。该项目的第二个研究重点将通过开发新的几何感知内核、采集函数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限。最后，研究人员将探索结合集成卡尔曼方法和贝叶斯优化的协同方法，利用这两个算法系列的优点来弥补它们的弱点。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进行评估，被认为值得支持以及更广泛的影响审查标准。