Approximating partial differential equations without boundary conditions

The predicting power of computational tools is of paramount importance in engineering and science. They offer insights into the behavior of complex systems, modeled by partial differential equations inside a region of interest. Boundary conditions expressing the influence of the surroundings must be provided to complete the mathematical models. However, there are many instances for which the boundary conditions are not available to practitioners: the understanding of the physical processes might be lacking, for instance when modeling the airflow around an airplane, or the boundary data is not accessible. This project aims to design numerical algorithms able to alleviate missing information on boundary conditions by incorporating physical measurements of the quantity of interest. The problems to be addressed fit under the strategic area of machine learning, and the potential scientific impact of this research is far-reaching. It includes improved meteorological forecasting, discovering biological pathways, and commercial design. In traditional numerical treatments of elliptic partial differential equations, the solution to be approximated is completely characterized by the given data. However, there are many instances for which the boundary conditions are not available. While not sufficient to pinpoint the solution, measurements of the solution are provided to attenuate the incomplete information. The aim of this research program is to exploit the structure provided by the PDE to design and analyze practical numerical algorithms able to construct the best simultaneous approximation of all functions satisfying the PDE and the measurements. This project embeds the design, analysis, and implementation of numerical methods for PDEs within an optimal recovery framework. It uncovers uncharted problematics requiring new mathematical tools. 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.