Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control
Abstract
Recently, efficient sweep-based low-rank iterative solvers have been developed to reduce the computational costs of steady-state solves and implicit time integration of the radiative transfer equation (RTE). In particular, angular sampling based on the discrete empirical interpolation method (DEIM) or on error-indicated randomized greedy selection enables efficient, non-intrusive reuse of existing transport sweep implementations. However, randomized-greedy low-rank source iteration with diffusion synthetic acceleration (SI--DSA) enforces uniformly tight accuracy tolerances across all source iterations, thereby oversolving the early iterations, and it does not explicitly reuse angular bases from preceding iterations. More critically, like its full-rank counterpart, this approach may lose effectiveness in multiscale problems with strong material discontinuities. We address these limitations through two complementary developments. First, we introduce adaptive accuracy control into randomized-greedy low-rank SI--DSA, reducing intermediate ranks and computational cost while maintaining the accuracy required for convergence. A two-stage hybrid sampling strategy can provide further acceleration by switching to DEIM-based sampling near convergence, once the preceding angular basis becomes sufficiently representative. Second, we develop an inexact flexible GMRES--DSA solver for the integral scalar-flux reformulation, in which sweep-based low-rank approximations are confined to the construction of the right-hand side and the matrix--vector products. The scalar-flux Krylov basis is not compressed, thereby preserving Arnoldi orthogonality in exact arithmetic and allowing established inexact-GMRES analysis to guide adaptive accuracy control of low-rank operations. We present numerical results on benchmark problems that demonstrate the effectiveness of the proposed methods.
Related
- On a boundary value problem related to radiative heat transfer
- A Robust High-order Approximation of the Gray Thermal Radiative Transfer Equations
- Dimension-Adaptive Accuracy Certificates for Convex Optimization
- Adaptive accuracy-driven splitting in Ozaki matrix multiplication and application to block LU factorization
- Structural equation modelling--based meta-analytic model for diagnostic test accuracy studies
BibTeX
@misc{han2026lowrank,
title = {{Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control}},
author = {Chenxi Han and Lukas Einkemmer and Wei Guo and Zhichao Peng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07653},
eprint = {2610.07653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07653},
}
RIS
TY - UNPB
TI - Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control
AU - Chenxi Han
AU - Lukas Einkemmer
AU - Wei Guo
AU - Zhichao Peng
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07653
UR - https://arxiv.org/abs/2610.07653
AB - Recently, efficient sweep-based low-rank iterative solvers have been developed to reduce the computational costs of steady-state solves and implicit time integration of the radiative transfer equation (RTE). In particular, angular sampling based on the discrete empirical interpolation method (DEIM) or on error-indicated randomized greedy selection enables efficient, non-intrusive reuse of existing transport sweep implementations. However, randomized-greedy low-rank source iteration with diffusion synthetic acceleration (SI--DSA) enforces uniformly tight accuracy tolerances across all source iterations, thereby oversolving the early iterations, and it does not explicitly reuse angular bases from preceding iterations. More critically, like its full-rank counterpart, this approach may lose effectiveness in multiscale problems with strong material discontinuities. We address these limitations through two complementary developments. First, we introduce adaptive accuracy control into randomized-greedy low-rank SI--DSA, reducing intermediate ranks and computational cost while maintaining the accuracy required for convergence. A two-stage hybrid sampling strategy can provide further acceleration by switching to DEIM-based sampling near convergence, once the preceding angular basis becomes sufficiently representative. Second, we develop an inexact flexible GMRES--DSA solver for the integral scalar-flux reformulation, in which sweep-based low-rank approximations are confined to the construction of the right-hand side and the matrix--vector products. The scalar-flux Krylov basis is not compressed, thereby preserving Arnoldi orthogonality in exact arithmetic and allowing established inexact-GMRES analysis to guide adaptive accuracy control of low-rank operations. We present numerical results on benchmark problems that demonstrate the effectiveness of the proposed methods.
ER -