In this work we propose an adaptive multilevel version of subset simulation to estimate the probability of rare events for complex physical systems. Given a sequence of nested failure domains of increasing size, the rare event probability is expressed as a product of conditional probabilities. The proposed new estimator uses different model resolutions and varying numbers of samples across the hierarchy of nested failure sets. In order to dramatically reduce the computational cost, we construct the intermediate failure sets such that only a small number of expensive high-resolution model evaluations are needed, whilst the majority of samples can be taken from inexpensive low-resolution simulations. A key idea in our new estimator is the use of a posteriori error estimators combined with a selective mesh refinement strategy to guarantee the critical subset property that may be violated when changing model resolution from one failure set to the next. The efficiency gains and the statistical properties of the estimator are investigated both theoretically via shaking transformations, as well as numerically. On a model problem from subsurface flow, the new multilevel estimator achieves gains of more than a factor 60 over standard subset simulation for a practically relevant relative error of 25%.

翻译：本文提出一种自适应多层子集模拟方法，用于估计复杂物理系统中稀有事件的概率。给定一系列规模递增的嵌套失效域，稀有事件概率被表示为条件概率的乘积。该新估计量在嵌套失效集层级中采用不同模型分辨率及变化样本数量。为显著降低计算成本，我们构建中间失效集，使得仅需少量昂贵的高分辨率模型评估，而大多数样本可从低成本的低分辨率模拟中获取。新估计量的核心思想是：结合后验误差估计量与选择性网格细化策略，以保障当模型分辨率在不同失效集间切换时可能被破坏的关键子集特性。通过抖动变换进行理论分析，并结合数值实验，研究了该估计量的效率提升与统计特性。针对地下水流模型问题，当相对误差为25%（实际工程相关精度）时，新多层估计量相比标准子集模拟实现了超过60倍的效率提升。