Algorithmic and analytical development for solving and learning nonlocal models

Nonlocal interactions have become increasingly prominent in natural systems, which has led to the growing interest in nonlocal modeling. The study of nonlocal models has attracted much attention from applied and computational mathematicians, motivated by applications in disciplines like mechanics, materials science, life science, data science, and social science. This project aims to study an important class of nonlocal models involving nonlocal interactions of a finite range. In a variety of real world applications, these models can serve as an alternative or complement to traditional PDE-based models and help connect PDE models with discrete models. The project is consistent with our vision of informative and intelligent scientific computing, which will broadly contribute to the advance of computation science and various scientific and engineering research fields, particularly as the world becomes increasingly connected (and nonlocal). The investigator will continue to build a close working relationship with research scientists at different institutions to facilitate the collaborative research effort and to strengthen the training and mentoring of young students and junior researchers. Effort will be made to ensure the timely translation and integration of new research findings into enhanced modeling and simulation capabilities for applications. Training at least one graduate student on the topics of the proposed research is expected.Nonlocal models differ from the more common local models represented by partial differential equations (PDEs) in that they employ nonlocal operators in integral forms, which can account for nonlocal interactions explicitly and provide more general modeling choices. Despite much progress, the mathematical theory and numerical analysis of nonlocal models are still at a nascent stage. Many key questions remain unanswered in all aspects of nonlocal modeling, analysis, and computation. In this project, the investigator aims to advance the mathematical and algorithmic development of nonlocal models with a finite range of interactions through an integrated solution and learning process. On one hand, the investigator will focus on a few key questions related to the formulation, discretization, and learning of nonlocal models, particularly involving physical boundaries or interfaces. This is represented by some specific tasks concerning the analytical and algorithmic development for solving nonlocal problems with inhomogeneous data at or near the boundary and/or with heterogeneous nonlocal interactions. These problems are notoriously challenging and their satisfactory solutions will have a significant impact on applications. Meanwhile, the project will help us build integrated modeling and learning processes. New strategies and algorithms will be developed to help bring potentially transformative changes to how nonlocal models are formulated and learned and to the related numerical methods, including their development, analysis, and application. By drawing close connections to local PDEs, fractional, and discrete graph models, the project will also shed light on the mathematical and computational studies of many other related subjectsThis 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.