现代蒙特卡罗变体用于中子输运的不确定性量化
Modern Monte Carlo Variants for Uncertainty Quantification in Neutron Transport
AI总结：
本文研究了中子输运方程在离散 ordinates 方法与 diamond 差分近似下的现代蒙特卡罗方法变体，探讨了单能一维 slab 几何问题中随机截面的不确定性量化方法及多级变体的效率提升。
摘要：
我们描述了用于中子输运方程不确定性量化的现代蒙特卡罗方法变体，当其通过离散 ordinates 方法与 diamond 差分近似进行近似时。我们聚焦于单能一维 slab 几何问题，其中散射是各向同性的，而截面是可能具有低正则性的对数正态相关随机场。本文包括关于离散方案收敛性的新理论结果概述，适用于空间变化和随机截面的情况。我们还描述了用于量化标量通量线性泛函不确定性的算法理论与实践，使用蒙特卡罗和拟蒙特卡罗方法及其多级变体。还提出了一个混合迭代/直接求解器，用于计算每个实现实的泛函。数值实验展示了混合求解器的有效性以及通过拟蒙特卡罗采样和多级方差减少所获得的收益。对于多级拟蒙特卡罗方法，我们观察到计算 ε 成本的收益可达标准蒙特卡罗方法的两个数量级，我们对此进行了理论解释。包括了具有多达数千个随机维度的问题实验。
英文摘要：
We describe modern variants of Monte Carlo methods for Uncertainty Quantification (UQ) of the Neutron Transport Equation, when it is approximated by the discrete ordinates method with diamond differencing. We focus on the mono-energetic 1D slab geometry problem, with isotropic scattering, where the cross-sections are log-normal correlated random fields of possibly low regularity. The paper includes an outline of novel theoretical results on the convergence of the discrete scheme, in the cases of both spatially variable and random cross-sections. We also describe the theory and practice of algorithms for quantifying the uncertainty of a linear functional of the scalar flux, using Monte Carlo and quasi-Monte Carlo methods, and their multilevel variants. A hybrid iterative/direct solver for computing each realisation of the functional is also presented. Numerical experiments show the effectiveness of the hybrid solver and the gains that are possible through quasi-Monte Carlo sampling and multilevel variance reduction. For the multilevel quasi-Monte Carlo method, we observe gains in the computational $\varepsilon$-cost of up to 2 orders of magnitude over the standard Monte Carlo method, and we explain this theoretically. Experiments on problems with up to several thousand stochastic dimensions are included.
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