辐射传输方程的低秩SI-DSA与GMRES-DSA方法及自适应精度控制
Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control
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- The Hong Kong University of Science and Technology（香港科技大学）
- University of Innsbruck（因斯布鲁克大学）
- Texas Tech University（德克萨斯理工大学）
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中文总结 AI 辅助
针对辐射传输方程低秩迭代求解器在早期迭代过求解及多尺度问题失效的缺陷，提出自适应精度控制与两阶段混合采样策略，并开发非精确柔性GMRES-DSA求解器，数值实验验证了其有效性。
中文摘要 AI 辅助
近年来，已开发出高效的基于扫描的低秩迭代求解器，以降低辐射传输方程（RTE）稳态求解和隐式时间积分的计算成本。特别是，基于离散经验插值法（DEIM）或基于误差指示的随机贪婪选择的角采样，能够高效、非侵入式地复用现有的输运扫描实现。然而，带有扩散合成加速的随机贪婪低秩源迭代（SI-DSA）在所有源迭代中强制执行均匀严格的精度容差，从而过度求解了早期迭代，并且它没有显式复用先前迭代的角基。更关键的是，与其满秩对应方法一样，该方法在具有强材料不连续性的多尺度问题中可能失效。我们通过两项互补的发展来解决这些局限性。首先，我们将自适应精度控制引入随机贪婪低秩SI-DSA，在保持收敛所需精度的同时，降低中间秩和计算成本。一种两阶段混合采样策略可以通过在收敛附近、当前角基已足够有代表性时切换到基于DEIM的采样来进一步加速。其次，我们为积分标量通量重构开发了一个非精确柔性GMRES-DSA求解器，其中基于扫描的低秩近似仅限于右端项和矩阵-向量乘积的构造。标量通量Krylov基不被压缩，从而在精确算术中保持Arnoldi正交性，并允许成熟的非精确GMRES分析来指导低秩操作的自适应精度控制。我们给出了基准问题的数值结果，证明了所提出方法的有效性。
英文摘要
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.
