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,用于准确估计高维方面难得事件或失败概率的精确估计。拟议方法结合了大偏差理论和适应性重要性抽样中的观点。重要取样员使用跨热带方法寻找最佳高山偏差分布,并重新使用整个过程为两者、目标概率估计和更新偏差分布而制作的所有样本。大偏差理论用于通过优化问题的解决方案找到良好的初始偏差分布。此外,它还用于确定一个低维次空间,这一次空间最能说明稀有事件概率。这一次空间用于跨热带方法,已知该方法在较高层面会丧失效率。拟议方法并不要求指标功能的平滑,也不涉及数字调整参数。我们用三个例子将该方法与最新水平的跨热带重要取样计划进行比较:一个高维概率估计基准,一个受扩散方程式制约的问题,以及一个由一个空间层面依赖时间的浅水系统管理的海啸问题。