A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation

Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 639-666, March 2024.
Abstract.Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high-dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there has been a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work toward that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput., 91 (2022), pp. 1–27] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed RBM (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: (1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and (2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux. We perform benchmark test problems in 2D2V, and the numerical tests show that the MMD-RBM can capture the low-rank structure effectively when it exists. A careful study in the computational cost shows that the offline stage of the MMD-RBM is more efficient than the proper orthogonal decomposition method, and in the low-rank case, it even outperforms a standard full-order solve. Therefore, the proposed MMD-RBM can be seen both as a surrogate builder and a low-rank solver at the same time. Furthermore, it can be readily incorporated into multiquery scenarios to accelerate problems arising from uncertainty quantification, control, inverse problems, and optimization.