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.
