Reduced Basis Methods for Parametric Steady-State Radiative Transfer Equation
Kimberly Matsuda, Yanlai Chen, Yingda Cheng, Fengyan Li
TL;DR
This work develops four projection-based reduced-order models within the reduced basis framework to efficiently solve the parametric steady-state radiative transfer equation with isotropic scattering and one energy group. By combining Galerkin and least-squares Petrov-Galerkin projections with L1 or residual error indicators, and enforcing an affine parameter dependence, the authors design robust offline-online strategies that yield substantial speedups (up to 10^4–10^6) in 2D2v problems. They provide rigorous complexity analyses, implement efficient residual evaluations, and demonstrate accurate, stagnation-free reduced surrogates across 1D and 2D2v benchmarks, including challenging lattice and pin-cell geometries. The results highlight the practicality of RBMs for parametric transport models and lay groundwork for certified ROMs in broader radiative transfer contexts.
Abstract
The radiative transfer equation (RTE) is a fundamental mathematical model to describe physical phenomena involving the propagation of radiation and its interactions with the host medium. Deterministic methods can produce accurate solutions without any statistical noise, yet often at a price of expensive computational costs originating from the intrinsic high dimensionality of the model. With this work, we present the first systematic investigation of projection-based reduced order models (ROMs) following the reduced basis method (RBM) framework to simulate the parametric steady-state RTE with isotropic scattering and one energy group. Four ROMs are designed, with each defining a nested family of reduced surrogate solvers of different resolution/fidelity. They are based on either a Galerkin or least-squares Petrov-Galerkin projection and utilize either an $L_1$ or residual-based importance/error indicator. Two of the proposed ROMs are certified in the setting when the absorption cross section is positively bounded below uniformly. One technical focus and contribution lie in the proposed implementation strategies under the affine assumption of the parameter dependence of the model. These well-crafted broadly applicable strategies not only ensure the efficiency and accuracy of the offline training stage and the online prediction of reduced surrogate solvers, they also take into account the conditioning of the reduced systems as well as the stagnation-free residual evaluation for numerical robustness. Computational complexities are derived for both the offline training and online prediction stages of the proposed model order reduction strategies, and they are demonstrated numerically along with the accuracy and robustness of the reduced surrogate solvers. Numerically we observe four to six orders of magnitude speedup of our ROMs compared to full order models for some 2D2v examples.