Highly Efficient Rank-Adaptive Sweep-based SI-DSA for the Radiative Transfer Equation via Mild Space Augmentation
This addresses computational bottlenecks in radiative transfer simulations for scientific computing, though it is incremental as it builds on existing low-rank and SI-DSA methods.
The paper tackled the inefficiency of rank-adaptive methods for the radiative transfer equation by developing a sweep-based solver with mild space augmentation, achieving accuracy and iteration counts comparable to full-rank methods while reducing memory usage and runtime, with effective ranks reaching 30-45% of full rank in challenging problems.
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.