Reduced Basis Methods for Parametric Steady-State Radiative Transfer Equation

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.

💡 Research Summary

This paper presents the first systematic investigation of projection‑based reduced order models (ROMs) for the parametric steady‑state radiative transfer equation (RTE) with isotropic scattering and a single energy group. The authors adopt a high‑fidelity full order model (FOM) based on an upwind discontinuous Galerkin (DG) spatial discretization (piecewise Q₁) coupled with the discrete ordinates (Sₙ) angular discretization. The resulting algebraic system is solved by source iteration accelerated with diffusion synthetic acceleration (SI‑DSA), a scheme known to be robust across a wide range of scattering and absorption regimes.

The reduced basis method (RBM) is employed to construct low‑dimensional surrogate spaces. A greedy algorithm iteratively enriches the reduced basis by selecting the parameter value at which the current reduced model exhibits the largest error, thereby requiring only a minimal number of expensive FOM solves. Four distinct ROMs are built by combining two projection strategies—standard Galerkin (G) and least‑squares Petrov‑Galerkin (PG)—with two error indicators: an L¹‑norm based indicator (L1) and a residual‑based indicator (Res). The four models are denoted G‑L1, G‑Res, PG‑Res, and PG‑L1.

Key technical contributions include:

1. Projection Choices – Galerkin projection yields a symmetric‑like reduced system when the underlying operator is self‑adjoint, while PG projection solves a least‑squares minimization of the residual, leading to better conditioning for highly non‑symmetric transport operators.

2. Error Indicators – The L¹ indicator provides a cheap, global measure of approximation quality and drives the greedy selection efficiently. The residual‑based indicator computes the actual operator residual, enabling rigorous a‑posteriori error bounds and certification of the reduced model when the absorption cross‑section is uniformly bounded below.

3. Affine Parameter Dependence – Assuming that material coefficients (total, scattering cross‑sections) and source terms depend affinely on the parameter vector μ, the authors pre‑assemble parameter‑independent matrices. During the online stage, the reduced system matrices and right‑hand sides are assembled by inexpensive linear combinations of these pre‑computed components.

4. Implementation Strategies – The paper introduces QR‑based algorithms (Algorithms 3‑5) for efficient residual minimization and evaluation without stagnation as the reduced dimension grows. Theoretical results (Theorems 1‑2) guarantee that the conditioning of the reduced systems remains moderate and that the residual evaluation is numerically stable.

5. Complexity Analysis – Offline, building an m‑dimensional ROM costs O(m · N) operations for the m FOM solves (N is the FOM dimension) plus O(m³) for reduced‑basis orthogonalization. Online, each query requires solving an m × m reduced system (O(m³)) and evaluating the residual (O(m·Q), where Q is the number of affine terms). Crucially, the online cost is independent of N.

6. Numerical Experiments – The authors test the four ROMs on 1‑D slab and 2‑D‑2v benchmark problems. For 2‑D‑2v cases with moderate FOM sizes (N≈10⁶), a reduced basis of dimension m = 15 yields speed‑ups of 10⁴–10⁶ while maintaining relative errors below 10⁻⁴. G‑Res and PG‑Res are certified under the absorption‑lower‑bound assumption; PG‑L1 shows sensitivity to the initial greedy selections but can be stabilized with an enhanced L¹ indicator.

Overall, the work demonstrates that RBM, when combined with careful projection choices, robust error indicators, and affine‑aware offline/online decomposition, provides a powerful, certified, and highly efficient surrogate for parametric transport problems. The methodology is broadly applicable to other high‑dimensional PDEs with affine parameter dependence and opens the door to fast multi‑query simulations in inverse problems, optimal design, and uncertainty quantification for radiative transfer and related transport phenomena.