SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 818-847, January 2021.
The estimation of rare event or failure probabilities in high dimensions is of interest in many areas of science and technology. We consider problems where the rare event is expressed in terms of a computationally costly numerical model. Importance sampling with the cross-entropy method offers an efficient way to address such problems provided that a suitable parametric family of biasing densities is employed. Although some existing parametric distribution families are designed to perform efficiently in high dimensions, their applicability within the cross-entropy method is limited to problems with dimension of $\mathcal{O}(10^2)$. In this work, rather than directly building sampling densities in high dimensions, we focus on identifying the intrinsic low-dimensional structure of the rare event simulation problem. To this end, we exploit a connection between rare event simulation and Bayesian inverse problems. This allows us to adapt dimension reduction techniques from Bayesian inference to construct new, effectively low-dimensional, biasing distributions within the cross-entropy method. In particular, we employ the approach in [O. Zahm et al., preprint, arXiv:1807.03712v2, 2018], as it enables control of the error in the approximation of the optimal biasing distribution. We illustrate our method using two standard high-dimensional reliability benchmark problems and one structural mechanics application involving random fields.

中文翻译：

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用于罕见事件模拟的基于交叉熵的重要性采样与故障通知降维

SIAM/ASA 不确定性量化杂志，第 9 卷，第 2 期，第 818-847 页，2021 年 1 月。
高维中罕见事件或故障概率的估计在科学和技术的许多领域都引起了人们的兴趣。我们考虑用计算成本高的数值模型来表示罕见事件的问题。使用交叉熵方法的重要性采样提供了一种解决此类问题的有效方法，前提是采用了合适的偏置密度参数族。尽管一些现有的参数分布族被设计为在高维度上高效执行，但它们在交叉熵方法中的适用性仅限于维度为 $\mathcal{O}(10^2)$ 的问题。在这项工作中，我们不是直接在高维中建立采样密度，而是专注于识别稀有事件模拟问题的内在低维结构。为此，我们利用罕见事件模拟和贝叶斯逆问题之间的联系。这使我们能够适应贝叶斯推理中的降维技术，以在交叉熵方法中构建新的、有效的低维偏置分布。特别是，我们采用了 [O. Zahm 等人，预印本，arXiv:1807.03712v2, 2018]，因为它可以控制最佳偏置分布的近似误差。我们使用两个标准的高维可靠性基准问题和一个涉及随机场的结构力学应用来说明我们的方法。特别是，我们采用了 [O. Zahm 等人，预印本，arXiv:1807.03712v2, 2018]，因为它可以控制最佳偏置分布的近似误差。我们使用两个标准的高维可靠性基准问题和一个涉及随机场的结构力学应用来说明我们的方法。特别是，我们采用了 [O. Zahm 等人，预印本，arXiv:1807.03712v2, 2018]，因为它可以控制最佳偏置分布的近似误差。我们使用两个标准的高维可靠性基准问题和一个涉及随机场的结构力学应用来说明我们的方法。