Highly Efficient Rank-Adaptive Sweep-based SI-DSA for the Radiative Transfer Equation via Mild Space Augmentation

TL;DR

This paper introduces a novel rank-adaptive sweep-based SI-DSA method for the radiative transfer equation that reduces memory and computational costs while maintaining accuracy, especially for complex multiscale problems.

Contribution

It develops a low-rank, sweep-based source iteration with diffusion synthetic acceleration that adaptively manages rank without excessive space augmentation, improving efficiency.

Findings

Achieves comparable accuracy and iteration counts to full-rank methods.

Substantially reduces memory usage and runtime.

Effective for multiscale problems with high effective rank.

Abstract

Low-rank methods have emerged as a promising strategy for reducing the memory footprint and computational cost of discrete-ordinates discretizations of the radiative transfer equation (RTE). However, most existing rank-adaptive approaches rely on rank-proportional space augmentation, which can negate efficiency gains when the effective solution rank becomes moderately large. To overcome this limitation, we develop a rank-adaptive sweep-based source iteration with diffusion synthetic acceleration (SI-DSA) for the first-order steady-state RTE. The core of our method is a sweep-based inner-loop iterative low-rank solver that performs efficient rank adaptation via mild space augmentation. In each inner iteration, the spatial basis is augmented with a small, rank-independent number of basis vectors without truncation, while a single truncation is performed only after the inner loop converges. Efficient rank adaptation is achieved through residual-based greedy angular subsampling strategy together with incremental updates of projection operators, enabling non-intrusive reuse of existing transport-sweep implementations. In the outer iteration, a DSA preconditioner is applied to accelerate convergence. Numerical experiments show that the proposed solver achieves accuracy and iteration counts comparable to those of full-rank SI-DSA while substantially reducing memory usage and runtime, even for challenging multiscale problems in which the effective rank reaches 30-45% of the full rank.