Multigroup Thermal Radiation Transport with Tensor Trains

TL;DR

This paper introduces a tensor-train (TT) based solver for multigroup thermal radiation transport (MRT), extending a previously gray TT method to include frequency groups by merging space and frequency into a single spatio-spectral TT core. Across a suite of challenging tests (Hohlraum, thermal relaxation with small and large $N_ u$, Gaussian diffusion, Graziani prompt spectrum with varying opacities, and stellar irradiation), the TT approach achieves substantial storage compression (often >$10^3$–$10^5$) and significant speedups (often >$10^2$), while preserving accuracy through energy conservation and to within prescribed tolerances. The study analyzes the internal structure of the merged core, defines internal ranks $r_{ u x}$ and $r_{x u}$, and compares TT topologies that separate space and frequency; results show that compression gains depend on problem regime, with certain problems favoring frequency- or space-angle-proximate decompositions. The work discusses binwise vs full multigroup TT representations, observes pointwise errors from rank truncation, and highlights practical trade-offs, suggesting avenues for future enhancements such as alternate TT topologies and randomized SVD techniques to further exploit low-rank structure in MRT. Overall, the method enables high-fidelity MRT computations with many frequency groups that were previously intractable on single nodes, offering a path toward scalable, non-gray radiative transfer simulations in complex physics.

Abstract

We investigate the application of tensor train (TT) algorithms to multigroup thermal radiation transport (i.e., photon radiation transport). The TT framework enables simulations at discretizations that might otherwise be computationally infeasible on conventional hardware. We show that solutions to certain multigroup problems possess an intrinsic low-rank structure, which the TT representation leverages effectively. This enables us to solve problems where the discretized solution size exceeds a trillion parameters on a single node. We consistently achieve compression factors $>$100$\times$ and speedups $>$2$\times$. The solver is evaluated across a range of test problems with varying levels of complexity. In addition, we further analyze the low-rank structure of the merged spatio-spectral core to evaluate the potential for additional compression via more advanced TT decompositions.

Figures (16)

• Figure 1: Graphical representation of the TT solution.
• Figure 2: Rank evolution of the multigroup hohlraum problem for both cases. The plot reveals that the rank evolution stays the same for both cases despite the discrepancy in initialization. We attribute this to the fact that the solution in every bin evolves exactly according to the gray solution scaled by the initial intensity in that bin. This leaves the TT-ranks unchanged for arbitrary intensity initializations across the frequency domain.
• Figure 3: Reduction of multigroup mean intensities to the gray case for case 1 (left) and 2 (center) plotted alongside the corresponding analytics (right) at $t=0.75$. Plots are overlayed with white contour lines at $\sum J_f = [0.1, 0.3, 0.5, 0.7]$. The numerical solution follows the analytical solution closely with compression and speedup amounting to $\sim$5150 and $\sim$73 respectively.
• Figure 4: Plot of the normalized radiaiton energy $\overline{E}_f$ v/s time. Different colored lines represent the solution evolution in different frequency groups and each plot converges to analytics as thermal equilibrium (at $T = T_{eq}$) is achieved.
• Figure 5: (a) Opacities $\kappa_{a,f}$ for the $N_f = 100$ relaxation test. (b) The variation of thermal energy density of the material ($\rho c_v T$) (blue), radiation energy density for some of the groups (colorbar), and the total radiation energy density (red) with time. The numerical solutions converge to the predicted analytic equilibrium. Notice that the radiation energy density generally reaches a steady value slower for groups with lower opacities (the darker lines flatten later compared to the orange ones).