Applications such as uncertainty quantification and optical tomography, require solving the radiative transfer equation (RTE) many times for various parameters. Efficient solvers for RTE are highly desired. Source Iteration with Synthetic Acceleration (SISA) is one of the most popular and successful iterative solvers for RTE. Synthetic Acceleration (SA) acts as a preconditioning step to accelerate the convergence of Source Iteration (SI). After each source iteration, classical SA strategies introduce a correction to the macroscopic particle density by solving a low order approximation to a kinetic correction equation. For example, Diffusion Synthetic Acceleration (DSA) uses the diffusion limit. However, these strategies may become less effective when the underlying low order approximations are not accurate enough. Furthermore, they do not exploit low rank structures concerning the parameters of parametric problems. To address these issues, we propose enhancing SISA with data-driven ROMs for the parametric problem and the corresponding kinetic correction equation. First, the ROM for the parametric problem can be utilized to obtain an improved initial guess. Second, the ROM for the kinetic correction equation can be utilized to design a low rank approximation to it. Unlike the diffusion limit, this ROM-based approximation builds on the kinetic description of the correction equation and leverages low rank structures concerning the parameters. We further introduce a novel SA strategy called ROMSAD. ROMSAD initially adopts our ROM-based approximation to exploit its greater efficiency in the early stage, and then automatically switches to DSA to leverage its robustness in the later stage. Additionally, we propose an approach to construct the ROM for the kinetic correction equation without directly solving it.

翻译：不确定性量化和光学断层扫描等应用需要针对不同参数多次求解辐射传输方程（RTE），因此亟需高效的RTE求解器。带合成加速的源迭代法（SISA）是RTE最流行且成功的迭代求解器之一。合成加速（SA）作为预处理步骤，用于加速源迭代（SI）的收敛。经典SA策略在每次源迭代后，通过求解动力学修正方程的低阶近似，引入宏观粒子密度的修正量。例如，扩散合成加速（DSA）采用扩散极限。然而，当底层低阶近似不够精确时，这些策略的效能可能下降。此外，它们未能利用参数化问题中与参数相关的低秩结构。为解决这些问题，我们提出利用数据驱动的ROM增强SISA，用于参数化问题及其对应的动力学修正方程。首先，参数化问题的ROM可用于获得改进的初始猜测。其次，动力学修正方程的ROM可用于设计其低秩近似。与扩散极限不同，这种基于ROM的近似建立在修正方程的动力学描述基础上，并利用了与参数相关的低秩结构。我们进一步提出一种新型SA策略——ROMSAD。ROMSAD在初始阶段采用基于ROM的近似以发挥其高效性，随后自动切换至DSA以利用其在后续阶段的鲁棒性。此外，我们还提出了一种无需直接求解动力学修正方程即可构建其ROM的方法。