NSF Award
2206762
The world is full of complex, multi-scale phenomena that can be challenging to predict due to their underlying chaotic nature. For example, fast and accurate predictions of weather phenomena (both terrestrial and solar), ocean dynamics, and groundwater flow are vital to economic growth and stability. These predictions typically incorporate computer simulations of mathematical models; however, to make accurate predictions, these models need to be properly "initialized"; that is, they need to know the current state of the system very precisely and the models need to be adjusted based on actual observations of the system in question. For example, in order to accurately predict the weather, models often require the current state of the weather to be known on the scale of a few inches, but weather observation stations are often spaced several miles apart. Since weather is a highly chaotic phenomenon, small errors in the observations and/or sparsity in the actual observations can lead to significant errors in the predictions. To address this issue in the past several decades a collection of techniques known as "data assimilation" have been developed. Data assimilation incorporates observational data into the mathematical model of the system of interest in order to drive the prediction to the correct state. However, the standard data assimilation techniques, known as the Kalman filter and four-dimensional variational (4D-VAR) approaches, are very computationally costly, and major challenges still exist when adapting them to complex systems. Recently, a new algorithm for data assimilation, known as the Azouani-Olson-Titi (AOT) algorithm has emerged as a fast, robust, highly accurate technique which is easy to adapt to a wide variety of models, and which is computationally inexpensive to add to an already existing computational model. This project will not only extend and improve the AOT algorithm, but it will also use new ideas and technologies invented by the PIs and coauthors to adapt the AOT framework to learn more about the underlying mathematical model itself, further improving predictive capabilities. This project fosters mentoring undergraduate and graduate students, interdisciplinary research, and interaction with national labs. The impacts of this project will be far-reaching and will pave the way for new techniques which will greatly speed up data assimilation in simulations of highly complicated fluid flows, introduce novel techniques for parameter learning and model reconstruction, and provide a computational approach to investigating fundamental mathematical problems. The computational technologies and mathematical tools developed will be useful to scientists and engineers in other fields as well.<br/><br/>This project builds on previous work of the PIs on the AOT algorithm, which showed that this algorithm can be adapted to learn the (unknown) parameters of the system, and even the form of the model itself, while simultaneously recovering the "true" state of the system. Extensions of the preliminary work will be carried out, and rigorous justification for convergence of the algorithm will be completed for physically interesting systems, including noisy data and sparse-in-time observations. In addition, several extensions of AOT itself will be numerically tested and rigorously investigated: nudging for intermittent observations, as well as nudging based on moving observers. AOT will also be implemented and tested for a multi-physics large-scale model of the Earth's oceans, for the Richards equation for soil moisture, and for a simplified fluids experiment using real-time collected data. This project will optimize observer requirements for better accuracy, significantly lowering costs. The methods described here have the potential to reduce production and computational cost for experiments, making them more useful to researchers working on real world problems. Moreover, novel proof methods will be developed to prove convergence in the cases of nonlinear AOT algorithms, AOT-based on moving observers, AOT-based model recovery, temperature-based AOT, and extensions of AOT to geophysical settings.<br/><br/>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.
