非光滑问题的自适应分层抽样
Adaptive stratified sampling for non-smooth problems
摘要：
不确定性的科学和工程问题通常都是计算昂贵的，并具有非光滑的参数依赖性，使标准蒙特卡罗太慢，并排除有效地使用加速不确定性量化方法依赖于严格的平滑假设。为了解决这些挑战，我们提出了一种自适应分层方法，适用于非光滑的问题，并显着降低方差相比，蒙特卡洛采样。分层迭代细化和样本顺序添加，以满足分配标准相结合的比例和最佳采样的好处。提供了预期性能和未能正确估计基本统计量的概率的理论估计。我们设计了一个实用的自适应分层方法，具有相同的几何形状，成本效益的细化满足贪婪的方差减少标准的层。数值实验证实了理论研究结果，并表现出高达三个数量级的加速比标准蒙特卡洛采样。
英文摘要：
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.
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