Tensor Networks for Block Adaptive Thermal Radiation Transport

Abstract

Thermal radiation transport (TRT) is computationally challenging because the photon specific intensity depends simultaneously on space, angle, and frequency, producing a high-dimensional state space whose cost grows rapidly with resolution. Tensor-network methods, such as the tensor train (TT) decomposition, can mitigate this curse of dimensionality when the solution admits low-rank structure, but localized nonseparable structure can drive substantial rank growth. Adaptive mesh refinement (AMR) provides a complementary means of making high spatial resolution tractable by concentrating resolution on features of interest that would otherwise be prohibitively expensive to resolve. When tensor representations are constructed locally on AMR blocks, this decomposition offers an additional advantage: nonseparable structure can remain confined to the blocks in which it occurs rather than driving rank growth throughout the domain. This paper introduces a framework for combining block-adaptive mesh refinement with block-local tensor-train representations of the photon specific intensity. The approach couples spatial adaptivity with localized low-rank compression, allowing fine-scale features to be resolved while limiting the impact of locally complex structure on tensor ranks elsewhere in the domain. We develop conservative ghost-cell transfer, coarse-fine flux correction, and refinement criteria that balance spatial discretization and tensor-rounding errors. Numerical experiments demonstrate that the method preserves low-rank structure across much of the domain while resolving localized features with AMR, achieving compression factors exceeding $\sim$10$^3$. The resulting framework provides a practical foundation for applying tensor-network methods to high-dimensional TRT on the multiscale, adaptively refined meshes encountered in radiation hydrodynamics simulations.