This study considers an efficient method for the estimation of quantiles associated to very small levels of probability (up to O(10−9)), where the scalar performance function J is complex (eg, output of an expensive-to-run finite element model), under a probability measure that can be recast as a multivariate standard Gaussian law using an isoprobabilistic transformation. A surrogate-based approach (Gaussian Processes) combined with adaptive experimental designs allows to iteratively increase the accuracy of the surrogate while keeping the overall number of J evaluations low. Direct use of Monte-Carlo simulation even on the surrogate model being too expensive, the key idea consists in using an importance sampling method based on an isotropic-centered Gaussian with large standard deviation permitting a cheap estimation of small quantiles based on the surrogate model. Similar to AK-MCS as presented in the work of Schobi et al., (2016), the surrogate is adaptively refined using a parallel infill criterion of an algorithm suitable for very small failure probability estimation. Additionally, a multi-quantile selection approach is developed, allowing to further exploit high-performance computing architectures. We illustrate the performances of the proposed method on several two to eight-dimensional cases. Accurate results are obtained with less than 100 evaluations of J on the considered benchmark cases.

中文翻译：

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使用自适应克里金法和重要性采样有效估计极端分位数

本研究考虑了一种估计与非常小的概率水平（高达 O(10−9)）相关的分位数的有效方法，其中标量性能函数 J 很复杂（例如，运行成本高昂的有限元的输出模型），在可以使用等概率变换重铸为多元标准高斯定律的概率度量下。基于代理的方法（高斯过程）与自适应实验设计相结合，可以迭代地提高代理的准确性，同时保持 J 评估的总数较低。即使在代理模型上直接使用蒙特卡罗模拟也太昂贵了，关键思想在于使用基于具有大标准偏差的各向同性中心高斯的重要性采样方法，允许基于代理模型对小分位数进行廉价估计。与 Schobi 等人 (2016) 的工作中提出的 AK-MCS 类似，代理使用适用于非常小的故障概率估计的算法的并行填充标准自适应地细化。此外，还开发了一种多分位数选择方法，允许进一步开发高性能计算架构。我们说明了所提出的方法在几个二维到八维情况下的性能。对所考虑的基准案例进行的 J 评估少于 100 次，即可获得准确的结果。使用适用于非常小的故障概率估计的算法的并行填充标准自适应地细化代理。此外，还开发了一种多分位数选择方法，允许进一步开发高性能计算架构。我们说明了所提出的方法在几个二维到八维情况下的性能。对所考虑的基准案例进行的 J 评估少于 100 次，即可获得准确的结果。使用适用于非常小的故障概率估计的算法的并行填充标准自适应地细化代理。此外，还开发了一种多分位数选择方法，允许进一步开发高性能计算架构。我们说明了所提出的方法在几个二维到八维情况下的性能。对所考虑的基准案例进行的 J 评估少于 100 次，即可获得准确的结果。