Efficient high-order methods for radiative transfer via substructuring

Many problems in science and engineering require predicting how radiation, light, or particles move through and interact with complex media. These problems arise in areas such as atmospheric science, optical imaging, nuclear engineering, and astrophysics, where accurate predictions are essential to scientific discovery, engineering design, and decision-making. A central tool for making such predictions is numerical simulation based on partial differential equations, which provides a first-principles way to model the relevant physical processes. However, these simulations remain very expensive because of the high-dimensional and multiscale nature of the underlying problems. This project aims to address this barrier by developing a systematic computational framework that integrates efficient high-order adaptive numerical methods, substructure-based parallel computation, and localized machine-learning models. The substructure-based design makes the computation well-suited for modern parallel computing systems, including high-performance computing clusters and GPUs. It also allows machine-learning models to be trained locally and inexpensively at the substructure level, while keeping the overall solver grounded in reliable and theoretically justified classical numerical methods. This combination of low-cost machine learning and first-principles-based numerical computation is expected to broaden access to the interdisciplinary area of scientific machine learning and provide students with training at the intersection of applied mathematics, high-performance computing, and artificial intelligence. This project will develop and analyze a unified substructuring framework for the radiative transfer equation. The research has three main goals. First, it will establish hp-explicit a priori error estimates for discontinuous Galerkin discretizations of the equation, clarifying how high-order schemes affect accuracy in both advection- and scattering-dominated regimes. Second, it will derive spectral estimates for statically condensed Schur complement systems, providing the theoretical foundation for substructuring solvers that are robust across varying mesh sizes, polynomial degrees, and scattering regimes. Third, the substructuring framework will enable reliable and inexpensive training of localized machine-learning models to accelerate the numerical scheme, while maintaining rigorous error control through perturbation analysis grounded in the spectral properties of the condensed system. Together, these components will contribute to an efficient, scalable, and theoretically grounded computational approach for radiative transfer and related high-dimensional transport problems. 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.