Thermal Radiation Transport with Tensor Trains

Abstract: We present a novel tensor network algorithm to solve the time-dependent, gray thermal radiation transport equation. The method invokes a tensor train (TT) decomposition for the specific intensity. The efficiency of this approach is dictated by the rank of the decomposition. When the solution is "low-rank," the memory footprint of the specific intensity solution vector may be significantly compressed. The algorithm, following a step-then-truncate approach of a traditional discrete ordinates method, operates directly on the compressed state vector thereby enabling large speedups for low-rank solutions. To achieve these speedups we rely on a recently developed rounding approach based on the Gram-SVD. We detail how familiar SN algorithms for (gray) thermal transport can be mapped to this TT framework and present several numerical examples testing both the optically thick and thin regimes. The TT framework finds low rank structure and supplies up to $\simeq$60$\times$ speedups and $\simeq$1000$\times$ compressions for problems demanding large angle counts, thereby enabling previously intractable SN calculations and supplying a promising avenue to mitigate ray effects.