Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature nonsmooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty quantification methods relying on strict smoothness assumptions. To remedy these challenges, we propose an adaptive stratification method suitable for nonsmooth problems and with significantly reduced variance compared to Monte Carlo sampling. The stratification is iteratively refined and samples are added sequentially to satisfy an allocation criterion combining the benefits of proportional and optimal sampling. Theoretical estimates are provided for the expected performance and probability of failure to correctly estimate essential statistics. We devise a practical adaptive stratification method with strata of the same kind of geometrical shapes, and cost-effective refinement satisfying a greedy variance reduction criterion. A Python implementation of the presented methodology is available at https://pypi.org/project/adaptive-stratification. Numerical experiments corroborate the theoretical findings and exhibit speedups of up to three orders of magnitude compared to standard Monte Carlo sampling.

中文翻译：

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非平滑问题的自适应分层抽样

受不确定性影响的科学和工程问题通常计算量大，并且具有非平滑参数依赖性，这使得标准蒙特卡罗太慢，并且不能有效使用依赖于严格平滑假设的加速不确定性量化方法。为了解决这些挑战，我们提出了一种适用于非光滑问题的自适应分层方法，与蒙特卡洛采样相比，方差显着降低。分层是迭代细化的，并按顺序添加样本以满足分配标准，该分配标准结合了比例抽样和最佳抽样的好处。为正确估计基本统计数据的预期性能和失败概率提供了理论估计。我们设计了一种实用的自适应分层方法，分层具有相同的几何形状，并且具有成本效益的细化满足贪婪方差减少标准。在 https://pypi.org/project/adaptive-stratification 中提供了所提出方法的 Python 实现。与标准蒙特卡罗采样相比，数值实验证实了理论发现并显示出高达三个数量级的加速。