This work introduces a novel framework for precisely and efficiently estimating rare event probabilities in complex, high-dimensional non-Gaussian spaces, building on our foundational Approximate Sampling Target with Post-processing Adjustment (ASTPA) approach. An unnormalized sampling target is first constructed and sampled, relaxing the optimal importance sampling distribution and appropriately designed for non-Gaussian spaces. Post-sampling, its normalizing constant is estimated using a stable inverse importance sampling procedure, employing an importance sampling density based on the already available samples. The sought probability is then computed based on the estimates evaluated in these two stages. The proposed estimator is theoretically analyzed, proving its unbiasedness and deriving its analytical coefficient of variation. To sample the constructed target, we resort to our developed Quasi-Newton mass preconditioned Hamiltonian MCMC (QNp-HMCMC) and we prove that it converges to the correct stationary target distribution. To avoid the challenging task of tuning the trajectory length in complex spaces, QNp-HMCMC is effectively utilized in this work with a single-step integration. We thus show the equivalence of QNp-HMCMC with single-step implementation to a unique and efficient preconditioned Metropolis-adjusted Langevin algorithm (MALA). An optimization approach is also leveraged to initiate QNp-HMCMC effectively, and the implementation of the developed framework in bounded spaces is eventually discussed. A series of diverse problems involving high dimensionality (several hundred inputs), strong nonlinearity, and non-Gaussianity is presented, showcasing the capabilities and efficiency of the suggested framework and demonstrating its advantages compared to relevant state-of-the-art sampling methods.

翻译：本文提出了一种新颖的框架，旨在精确高效地估计复杂高维非高斯空间中的罕见事件概率。该框架建立在我们先前提出的近似采样目标后处理调整（ASTPA）方法基础之上。首先构建并采样一个未归一化的采样目标，该目标放宽了最优重要性采样分布的要求，并专门针对非高斯空间进行设计。采样后，通过稳定的逆重要性采样程序估计其归一化常数，该程序采用基于已有样本构建的重要性采样密度。随后基于这两个阶段得到的估计值计算目标概率。我们对所提出的估计量进行了理论分析，证明了其无偏性并推导出其解析变异系数。为了对构建的目标进行采样，我们采用自主开发的拟牛顿质量预处理哈密顿蒙特卡洛方法（QNp-HMCMC），并证明其能收敛到正确的平稳目标分布。为避免在复杂空间中调整轨迹长度这一挑战性任务，本工作中有效采用了单步积分的QNp-HMCMC实现。由此我们证明了单步实现的QNp-HMCMC等价于一种独特高效的预处理Metropolis调整Langevin算法（MALA）。本文还利用优化方法有效初始化QNp-HMCMC，并最终讨论了所开发框架在有界空间中的实现。通过一系列涉及高维度（数百个输入）、强非线性和非高斯特性的多样化问题，展示了所提出框架的性能与效率，并证明了其相较于相关先进采样方法的优势。