We propose a method for the accurate estimation of rare event or failure probabilities for expensive-to-evaluate numerical models in high dimensions. The proposed approach combines ideas from large deviation theory and adaptive importance sampling. The importance sampler uses a cross-entropy method to find an optimal Gaussian biasing distribution, and reuses all samples made throughout the process for both, the target probability estimation and for updating the biasing distributions. Large deviation theory is used to find a good initial biasing distribution through the solution of an optimization problem. Additionally, it is used to identify a low-dimensional subspace that is most informative of the rare event probability. This subspace is used for the cross-entropy method, which is known to lose efficiency in higher dimensions. The proposed method does not require smoothing of indicator functions nor does it involve numerical tuning parameters. We compare the method with a state-of-the-art cross-entropy-based importance sampling scheme using three examples: a high-dimensional failure probability estimation benchmark, a problem governed by a diffusion equation, and a tsunami problem governed by the time-dependent shallow water system in one spatial dimension.

翻译：我们提出了一种方法，用于在高维场景下精确估计计算成本高昂的数值模型中的罕见事件或失效概率。该方法融合了大偏差理论与自适应重要抽样的思想。其重要抽样器采用交叉熵方法寻找最优高斯偏置分布，并在整个过程中重复利用所有样本，既用于目标概率估计，也用于更新偏置分布。大偏差理论通过求解优化问题来获得良好的初始偏置分布，同时用于识别最能反映罕见事件概率的低维子空间。该子空间被引入交叉熵方法，以缓解该方法在高维场景下效率降低的问题。所提方法无需对指示函数进行平滑处理，也不涉及数值调参。我们通过三个算例将其与当前最先进的基于交叉熵的重要抽样方案进行对比：高维失效概率估计基准测试、扩散方程控制问题，以及一维空间中由时变浅水系统控制的Tsunami问题。