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 a popular and successful iterative solver 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 novel SA strategy called ROMSAD. In the early stage, ROMSAD adopts a ROM based approximation, which builds on the kinetic description of the correction equation and leverages low rank structures concerning the parameters. This ROM-based approximation has greater efficiency than DSA in the early stage of SI. In the later stage, ROMSAD automatically switches to DSA to leverage its robustness. Additionally, we propose an approach to construct the ROM for the kinetic correction equation without directly solving it. In a sires of numerical tests, we compare the performance of the proposed methods with SI-DSA and DSA preconditioned GMRES solver.

翻译：在不确定性量化和光学层析成像等应用中，需要针对不同参数多次求解辐射传输方程（RTE）。因此，高效的RTE求解器备受关注。源迭代与合成加速（SISA）是一种流行且成功的RTE迭代求解器。合成加速（SA）作为预处理步骤，用于加速源迭代（SI）的收敛。在每次源迭代后，经典的SA策略通过求解动力学修正方程的低阶近似来对宏观粒子密度进行修正。例如，扩散合成加速（DSA）利用了扩散极限。然而，当基础的低阶近似不够精确时，这些策略可能效果不佳。此外，它们未能利用参数化问题中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降阶模型（ROM）来增强SISA，分别针对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可用于设计一种新型的SA策略，称为ROMSAD。在早期阶段，ROMSAD采用基于ROM的近似，该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。在SI的早期阶段，这种基于ROM的近似比DSA具有更高的效率。在后期阶段，ROMSAD自动切换至DSA以利用其鲁棒性。此外，我们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。在一系列数值测试中，我们将所提方法的性能与SI-DSA及DSA预处理的GMRES求解器进行了比较。