Radiative transfer equation (RTE) models particles propagating through and interacting with a background medium. Applications, such as uncertainty quantification, medical imaging, and shape optimization, require solving RTE many times for various parameters. Source Iteration (SI) with Diffusion Synthetic Acceleration (DSA) is a popular iterative solver for RTE. DSA can be seen as a preconditioning step to accelerate the convergence. DSA is based on the diffusion limit of a kinetic correction equation. However, when the underlying problem is far from its diffusion limit, DSA may become less effective. Furthermore, DSA does not exploit low-rank structures of the solution manifold concerning parameters of parametric problems. To address these issues, we enhance SI with data-driven reduced-order models. These data-driven ROMs still build on the original kinetic description of the correct equation and leverage low rank structures concerning parameters of parametric problems. A new preconditioner, which exploits the advantages of both data-driven ROMs and the classical DSA method, is developed.

Zhichao Peng is currently an assistant professor at the Department of Mathematics, the Hong Kong University of Science and Technology (HKUST). Before joining HKUST in summer 2023, he was a postdoc at the department of Mathematics at Michigan State University. Zhichao obtained his doctoral degree from Rensselaer Polytechnic Institute in 2020 and his bachelor degree from Peking University. His main research interests include numerical methods for kinetic equations and wave equations, and data-driven dimensionality reduction techniques.