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, cost-effective refinement satisfying a greedy variance reduction criterion. Numerical experiments corroborate the theoretical findings and exhibit speedups of up to three orders of magnitude compared to standard Monte Carlo sampling.

翻译：受不确定因素影响的科学和工程问题往往在计算上费用昂贵,而且具有非色谱参数依赖性的特点,使得标准蒙特卡洛太慢,并且排除了根据严格的顺畅假设有效使用加速的不确定性量化方法。为了克服这些挑战,我们建议一种适应性分层方法,适合非色问题,与蒙特卡洛抽样相比差异大为减少。分层是迭接式的,并按顺序添加样本,以满足分配标准,将比例和最佳抽样的好处结合起来。对预期性能和无法正确估计基本统计数据的可能性提供了理论估计。我们设计了一种实用的适应性分层方法,具有同样的几何形状,具有成本效益的完善方法,符合贪婪差异减少标准。数字实验证实了理论结果和显示的加速度,比标准蒙特卡洛抽样高出了三个数量级。