Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the asymptotic convergence rate is known, predicting the actual value of the statistical error remains a challenging problem. In this paper, we show how the statistical error of an analog particle tracing Monte Carlo method can be calculated (expensive) and predicted a priori (cheap) when estimating quantities of interest (QoI) on a histogram. We consider two types of QoI estimators: point estimators for which each particle provides one independent contribution to the QoI estimates, and analog estimators for which each particle provides multiple correlated contributions to the QoI estimates. The developed statistical error predictors can be applied to other QoI estimators and nonanalog simulation routines as well. The error analysis is based on interpreting the number of particle visits to a histogram bin as the result of a (correlated) binomial experiment. The resulting expressions can be used to optimize (non)analog particle tracing Monte Carlo methods and hybrid simulation methods involving a Monte Carlo component, as well as to select an optimal particle tracing Monte Carlo method from several available options. Additionally, the cheap statistical error predictors can be used to determine a priori the number of particles N that is needed to reach a desired accuracy. We illustrate the theory using a linear kinetic equation describing neutral particles in the plasma edge of a fusion device and show numerical results. The code used to perform the numerical experiments is openly available.

翻译：大型粒子系统通常由高维（线性）动力学方程描述，这些方程采用蒙特卡罗方法进行模拟，其渐近收敛速度与维度无关。尽管已知渐近收敛速度，但预测统计误差的实际值仍是一个具有挑战性的问题。本文展示了如何在直方图上估计感兴趣量（QoI）时，计算（昂贵）并先验预测（廉价）模拟粒子追踪蒙特卡罗方法的统计误差。我们考虑两类QoI估计器：点估计器（每个粒子对QoI估计提供一次独立贡献）和模拟估计器（每个粒子对QoI估计提供多次相关贡献）。所开发的统计误差预测器也可应用于其他QoI估计器和非模拟仿真程序。误差分析基于将粒子对直方图区间的访问次数解释为（相关）二项试验的结果。所得表达式可用于优化（非）模拟粒子追踪蒙特卡罗方法及含蒙特卡罗组件的混合模拟方法，也可用于从多个可选方案中选取最优粒子追踪蒙特卡罗方法。此外，廉价统计误差预测器可用于先验确定达到目标精度所需的粒子数N。我们以描述聚变装置等离子体边缘中性粒子的线性动力学方程为例阐释该理论，并展示数值结果。用于数值实验的代码已公开。