采用随机化 Halton 序列的迭代拟蒙特卡罗方法的批量幂迭代方法
A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence
AI总结：
本文提出基于随机化 Halton 序列的 RQMC 批量幂迭代 iQMC 方法，改善相空间采样并消除固定种子导致的射线效应，在 MC/DC 中实现，以更少粒子历史获得稳定、精确且 O(N^{-1}) 收敛的结果。
摘要：
迭代拟蒙特卡罗（iQMC）方法是近期开发的用于中子输运模拟的混合方法。iQMC 用拟蒙特卡罗模拟替代确定性线性求解器中使用的标准求积技术，以准确且高效地求解中子输运方程。以往的 iQMC 研究采用固定种子方法，即在每次输运扫描开始时，粒子都被重置到相同的初始位置和运动方向。尽管 QMC 样本相比伪随机样本提供了大幅改善的均匀性，但固定种子方法意味着问题的某些区域采样不足，并导致了与离散纵标法中观察到的射线效应相似的误差。本工作探索使用随机化拟蒙特卡罗技术（RQMC）为每次输运扫描生成独特的 QMC 样本集合，从而显著改善相空间采样。RQMC 的使用给 iQMC 的迭代过程引入了此前不存在的一些随机噪声。为弥补这一点，我们采用类似于典型蒙特卡罗 k 特征值问题的“批量”方法：iQMC 源在 N_inactive 个批量上收敛，随后记录 N_active 个批量的结果，并用于计算解的平均值和标准差。RQMC 批量方法在 Monte Carlo Dynamic Code（MC/DC）中实现，并被证明相较固定种子方法有大幅改进。该批量方法能够提供迭代稳定且更准确的解，同时每批粒子历史数减少了近两个数量级。值得注意的是，尽管给解引入了一些随机噪声，RQMC 批量方法对 k-effective 和平均标量通量误差均以 O(N^{-1}) 的理论 QMC 收敛速率收敛。
英文摘要：
The Iterative Quasi-Monte Carlo (iQMC) method is a recently developed hybrid method for neutron transport simulations. iQMC replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation for accurate and efficient solutions to the neutron transport equation. Previous iQMC studies utilized a fixed-seed approach wherein particles were reset to the same initial position and direction of travel at the start of every transport sweep. While the QMC samples offered greatly improved uniformity compared to pseudo-random samples, the fixed-seed approach meant that some regions of the problem were under-sampled and resulted in errors similar to ray effects observed in discrete ordinates methods. This work explores using randomized-Quasi Monte Carlo techniques (RQMC) to generate unique sets of QMC samples for each transport sweep and gain a much-improved sampling of the phase space. The use of RQMC introduces some stochastic noise to iQMC's iterative process, which was previously absent. To compensate, we adopt a ``batch'' approach similar to typical Monte Carlo k-eigenvalue problems, where the iQMC source is converged over $N_\text{inactive}$ batches, then results from $N_\text{active}$ batches are recorded and used to calculate the average and standard deviation of the solution. The RQMC batch method was implemented in the Monte Carlo Dynamic Code (MC/DC) and is shown to be a large improvement over the fixed-seed method. The batch method was able to provide iteratively stable and more accurate solutions with nearly two orders of magnitude reduction in the number of particle histories per batch. Notably, despite introducing some stochastic noise to the solution, the RQMC batch approach converges both the k-effective and mean scalar flux error at the theoretical QMC convergence rate of $O(N^{-1})$.
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