Feature-Based Data Assimilation and Uncertainty Quantification for Complex Systems in Science and Engineering

数据同化的基本思想是用来自稀疏和噪声数据的信息更新计算模型，使更新后的模型能够用于预测。数据同化是计算地球物理的核心，尤其是在数值天气预报、海洋学和地磁中，并广泛应用于工程应用，从机器人到油藏建模。在通常的方法中，人们试图改进计算模型，使其输出与数据匹配。然而，将模型输出直接匹配到数据通常是不必要的，甚至是不可取的。在这个项目中，数据同化得到了扩展，这样计算模型就可以基于数据中的特征而不是原始数据本身来更新。特征方法降低了内在维度，适用于地球科学和工程中的大规模问题，具体应用于地磁偶极子反转、云建模和太阳能电池的不确定性量化。该项目的主要技术目标是扩展数据同化，以便计算模型可以根据数据中观察到的特征而不是原始数据进行校准。这可以在贝叶斯框架内通过用从数据中计算出的合适的低维特征替换数据来实现。由此产生的基于特征的似然可以用来将精细尺度数据的选定方面吸收到粗糙的低维模型中。更一般地说，特征的使用降低了似然的维数，这反过来又降低了蒙特卡罗方法基于特征的数据同化的计算需求。基于特征的方法的数学基础将通过严格的分析来探索。将创建基于特征的数据同化的新计算方法，将机器学习技术与蒙特卡罗采样相结合。这些方法的效率将通过与地球科学和工程领域的科学家在三个具体应用领域的跨学科合作来评估。具体而言，将开发基于特征的数据同化算法，用于研究地球磁偶极子场的超时线，确定低维云模型的地球物理相关性，以及用于太阳能发电的薄膜聚合物反射器的不确定性量化。这些应用程序将协同连接多个学科（地球科学、工程、数学）的科学家（教师、博士后和学生）。亚利桑那大学的本科生和研究生将作为项目的一部分接受培训，并将帮助产生和传播关键成果。这些研究活动将伴随着一项外展计划，作为亚利桑那大学数学系G-Teams项目的一部分实施。推广活动的中心主题是通过将数学概念应用于与我们社会相关的问题，向K-12教师及其学生展示数学“在行动”。

The basic idea of data assimilation is to update a computational model with information from sparse and noisy data so that the updated model can be used for predictions. Data assimilation is at the core of computational geophysics, most notably in numerical weather prediction, oceanography, and geomagnetism, and is used widely in engineering applications, ranging from robotics to reservoir modeling. In the usual approach one attempts to refine a computational model such that its outputs match data. However, matching model outputs directly to data is often unnecessary or even undesirable. In this project, data assimilation is extended so that computational models can be updated based on features in the data, rather than the raw data themselves. The feature approach reduces an intrinsic dimension and is applicable to large scale problems in geosciences and engineering, with specific applications in geomagnetic dipole reversals, cloud modeling, and uncertainty quantification for solar cells.The primary technical aim of this project is to extend data assimilation such that computational models can be calibrated against features observed in the data, rather than the raw data. This can be achieved within a Bayesian framework by replacing the data with a suitable low-dimensional feature, computed from the data. The resulting feature-based likelihood can be used to assimilate selected aspects of fine-scale data into coarse, low-dimensional models. More generally, the use of features reduces the dimension of the likelihood, which in turn reduces the computational requirements of feature-based data assimilation by Monte Carlo methods. The mathematical foundations of the feature-based approach will be explored by rigorous analysis. New computational methods for feature-based data assimilation will be created, which combine machine learning techniques with Monte Carlo sampling. The efficiency of these methods will be assessed by interdisciplinary collaboration with scientists in geosciences and engineering in three specific applications. Specifically, feature-based data assimilation algorithms will be developed for the study of superchrons of Earth's magnetic dipole field, to determine the geophysical relevance of low-dimensional cloud models, and for uncertainty quantification of thin-film polymeric reflectors for solar power generation. These applications will collaboratively connect scientists (faculty, postdocs and students) across several disciplines (geosciences, engineering, mathematics). Undergraduate and graduate students at the University of Arizona will be trained as part of the project and will aid in producing and disseminating key results. The research activities will be accompanied by an outreach plan, implemented as part of the G-Teams program within the Department of Mathematics at the University of Arizona. A central outreach theme is to demonstrate, for K-12 teachers and their students, mathematics "in action" by applying mathematical concepts to problems relevant to our society.