This paper presents a robust version of the stratified sampling method when multiple uncertain input models are considered for stochastic simulation. Various variance reduction techniques have demonstrated their superior performance in accelerating simulation processes. Nevertheless, they often use a single input model and further assume that the input model is exactly known and fixed. We consider more general cases in which it is necessary to assess a simulation's response to a variety of input models, such as when evaluating the reliability of wind turbines under nonstationary wind conditions or the operation of a service system when the distribution of customer inter-arrival time is heterogeneous at different times. Moreover, the estimation variance may be considerably impacted by uncertainty in input models. To address such nonstationary and uncertain input models, we offer a distributionally robust (DR) stratified sampling approach with the goal of minimizing the maximum of worst-case estimator variances among plausible but uncertain input models. Specifically, we devise a bi-level optimization framework for formulating DR stochastic problems with different ambiguity set designs, based on the $L_2$-norm, 1-Wasserstein distance, parametric family of distributions, and distribution moments. In order to cope with the non-convexity of objective function, we present a solution approach that uses Bayesian optimization. Numerical experiments and the wind turbine case study demonstrate the robustness of the proposed approach.

翻译：本文针对随机模拟中考虑多个不确定输入模型的情形，提出了一种鲁棒版本的分层抽样方法。各类方差缩减技术在加速模拟过程中已展现出卓越性能，然而它们通常仅采用单一输入模型，且假设该模型是精确已知且固定不变的。我们考虑了更一般的情况——例如评估非平稳风况下风力涡轮机的可靠性，或客户到达间隔时间分布随时间异质的服务系统运行时——需要评估模拟结果对不同输入模型的响应。此外，输入模型的不确定性可能显著影响估计方差。为应对此类非平稳且不确定的输入模型，我们提出了一种分布鲁棒分层抽样方法，其目标是在多个可能的但不确定的输入模型中最小化最坏情况下的估计方差最大值。具体而言，我们设计了一个双层优化框架，基于$L_2$范数、1-Wasserstein距离、参数化分布族及分布矩等不同模糊集设定，来构建分布鲁棒随机问题。为处理目标函数的非凸性，我们提出了一种基于贝叶斯优化的求解方法。数值实验与风力涡轮机案例研究验证了所提方法的鲁棒性。