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, and it arises in many applications. 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. This is more prominent in multi-query tasks, e.g., inverse problems and optimal design, when the RTE needs to be solved repeatedly. This motivates the developments of dimensionality and model order reduction techniques for such transport models.
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. The use of RBM compared to standard proper orthogonal decomposition (POD) is well motivated, especially considering that a large number of degrees of freedom is needed by full order models to solve high dimensional transport models like RTE. 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 L1 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.