SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 2, Page 511-538, January 2020.
This paper deals with the estimation of rare event probabilities using importance sampling (IS), where an optimal proposal distribution is computed with the cross-entropy (CE) method. Although IS optimized with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of a partial differential equation (PDE) with random inputs. This work proposes to alleviate computation by the parsimonious use of a hierarchy of score function approximations in the CE optimization process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimization stage. The selection relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the method in the context of pollution alerts and a system modeled by a PDE.

中文翻译：

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在交叉熵方法中选择简化模型

SIAM / ASA不确定性量化期刊，第8卷，第2期，第511-538页，2020年1月。
本文使用重要性抽样（IS）来处理稀有事件概率的估计，其中使用交叉熵（CE）方法计算最佳建议分布。尽管使用CE方法优化的IS可以有效地减少估计量方差，但是对于重复计算得分函数表示过于费力的计算的问题，这种方法仍然难以承受。对于与具有随机输入的偏微分方程（PDE）的解相关的得分函数，通常是这种情况。这项工作提出通过在CE优化过程中简化使用评分函数近似值的层次结构来减轻计算量。得分函数近似值是通过选择最低维数来获得的，其准确性保证可以通过当前的CE优化阶段。选择取决于错误规范上的认证上限。渐近分析为所提算法的效率和收敛性提供了理论上的保证。数值结果证明了该方法在污染预警和PDE建模系统中带来的收益。