一维粒子输运问题中多层级混合蒙特卡洛方法的收敛性
Convergence of Multi-Level Hybrid Monte Carlo Methods for 1-D Particle Transport Problems
AI总结：
本文提出一种混合多层级蒙特卡洛方法求解中性粒子输运方程，通过粗网格基础解与修正因子降低方差和成本，并在1D平板模型中验证了弱收敛性。
摘要：
本文提出了一种用于求解中性粒子输运方程的混合多层级蒙特卡洛（MLMC）方法。MLMC方法最初是为了解决参数积分问题而开发的，其工作原理是使用廉价、低保真度的解作为基础解，然后在一系列计算网格上求解加性修正因子。所提出的算法通过基于低阶准扩散方程的混合蒙特卡洛方法生成标量通量样本。我们在每个网格上生成初始数量的样本，然后利用MLMC理论计算在每个层级上执行的最优样本数量。针对一维平板模型的计算结果展示了所考虑泛函的弱收敛性。所分析的泛函是标量通量解在整个域或特定子区域上的积分。我们观察到修正因子的方差下降速度快于生成MLMC样本成本的增长速度。MLMC解的方差和成本由粗网格计算主导。因此，我们应该能够以最小的成本增加额外的计算层级，因为在后续层级上收敛修正因子的估计所需的样本数量会更少。
英文摘要：
We present in this paper a hybrid, Multi-Level Monte Carlo (MLMC) method for solving the neutral particle transport equation. MLMC methods, originally developed to solve parametric integration problems, work by using a cheap, low fidelity solution as a base solution and then solves for additive correction factors on a sequence of computational grids. The proposed algorithm works by generating a scalar flux sample using a Hybrid Monte Carlo method based on the low-order Quasidiffusion equations. We generate an initial number of samples on each grid and then calculate the optimal number of samples to perform on each level using MLMC theory. Computational results are shown for a 1-D slab model to demonstrate the weak convergence of considered functionals. The analyzed functionals are integrals of the scalar flux solution over either the whole domain or over a specific subregion. We observe the variance of the correction factors decreases faster than increase in the cost of generating a MLMC sample grows. The variance and costs of the MLMC solution are driven by the coarse grid calculations. Therefore, we should be able to add additional computational levels at minimal cost since fewer samples would be needed to converge estimates of the correction factors on subsequent levels.
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