A priori rank structure in tensor-train radiative transfer: from quantization asymmetry to spectral universality

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Abstract
Tensor-train methods compress transport equations by orders of magnitude in benchmark calculations, but they largely lack a predictive theory: reported rank structures are empirical, measured a posteriori on specific problem instances, and no failure modes of quantized tensor-train (QTT) decomposition have been systematically identified for transport problems. This work develops a predictive framework for tensor-train rank behavior using two-dimensional radiative transfer with spectrally resolved opacity as a vehicle. We report three principal findings. First, we identify a fundamental operator--solution rank asymmetry under uniform QTT decomposition: spatial operator ranks remain bounded, but spatial solution ranks inflate as O(Nx) because the multivariate coupling of the transport solution lacks the algebraic separability that makes QTT effective for operators. Selective quantization, applying QTT only to the spectral dimension, resolves this asymmetry and achieves O(log Nσ) storage without spatial penalty. Second, combining this approach with Young-measure homogenization, we establish the first a priori spectral rank guarantee for tensor methods in radiative transfer equations: the spectral band rank is universally bounded at 2--3, traced to the two-mode structure of the exponential attenuation factor. Because the spectral and spatial--angular compression mechanisms act on disjoint tensor modes, their ratios compose multiplicatively, exceeding 104 at production scales. Third, the framework is validated on self-consistent collisional-radiative transport with atomic plasma opacity, where it closes the multiphysics coupling loop with six-digit accuracy---further demonstrating that tensor-compressed solvers can support real multiphysics workflows. The present work helps establish tensor-train methods for transport equations as not just a compression technique but a framework whose effectiveness can be predicted from the structure of the underlying physics.

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