Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The developed techniques in this paper are applicable from low- to high-dimensional stochastic spaces, and the basic idea is to construct a relevant target distribution by weighting the original random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.

翻译：精确高效地估计稀有事件概率具有重要意义，因为此类事件的发生往往会产生广泛影响。本文聚焦于精确量化这些在复杂工程系统可靠性分析中常见的概率，基于所提出的框架——近似采样目标与后处理调整（ASTPA），该框架在此与基于梯度的哈密顿马尔可夫链蒙特卡洛（HMCMC）方法相结合并得到支持。本文发展的技术适用于从低维到高维的随机空间，其基本思想是通过一维输出似然模型对原始随机变量空间进行加权，利用极限状态函数构建相关目标分布。为从该目标分布中采样，我们采用HMCMC算法——一类利用物理系统动力学而非单纯依赖提议概率分布来生成远距离序列样本的MCMC方法，并发展了一种新的拟牛顿质量预条件HMCMC方案（QNp-HMCMC），该方案特别适用于高维空间且效率显著。为最终计算稀有事件概率，我们设计了一个基于已有样本的逆重要性采样后处理步骤。本文还分析了估计量的统计性质，并通过一系列具有挑战性的低维与高维问题，详细检验了所提方法的性能，并将其与子集模拟方法进行了比较。