Radiative transfer equations (RTEs), as a class of kinetic equations, are fundamental models to describe the physical phenomena of energy or particle transport through mediums that are affected by absorption, emission, and scattering processes. Deterministic simulation can provide an accurate description of the solutions, however they face many computational challenges, most prominently the need to compute the angular flux which is defined in the high dimensional phase space. Leveraging the existence of a low-rank structure in the solution manifold induced by the angular variable in the scattering dominating regime, in this work, we design and test reduced order models (ROMs) for the linear RTE model based on reduced basis methods (RBMs). As an established ROM for parametric models, RBM features a greedy algorithm in the offline (i.e. training) stage. It selects a number of representative parameter values via a greedy procedure. During the online stage, a reduced solution for a given parameter is sought in the terminal surrogate space. The novel perspective we adopt for the RTE model is to treat the angular variable (also the time variable for the time-dependent case) as a parameter. Unlike standard parametric models, the solutions for different angular values are coupled for the RTE through the scattering operator which is an integral operator in the angular variable. Moreover, the unstructured nature of the selected angular samples prevents a direct numerical integration for computing the scattering operator robustly and accurately. Strategies are proposed to tackle these challenges within the RBM framework for both the stationary and time-dependent RTE models. This is a joint work with Y. Chen (UMass Dartmouth), Y. Cheng and Z. Peng (Michigan State Univ.).

Fengyan Li is a Professor in Applied Mathematics at Rensselaer Polytechnic Institute. She received her BS and MS degrees from Peking University, and her PhD degree from Brown University. Her research focuses on the design and analysis of robust, accurate and efficient computational methods, especially discontinuous Galerkin finite element methods, such as for ideal MHD equations, Maxwell's equations in linear or nonlinear media, kinetic models in rarefied gas dynamics and plasma physics (e.g. BGK, RTE, Vlasov-type equations).