This paper considers the classical problem of sampling with Monte Carlo methods a target probability distribution obtained by conditioning on a rare event defined by the level set of a real-valued score function that is very expensive to compute. We also consider a context where, with each new evaluation of the true score function, a method that iteratively builds a sequence of reduced scores is available; these reduced scores being moreover certified with pointwise error bounds. This work proposes a fully adaptive algorithm that iteratively: i) builds a sequence of proposal distributions obtained by conditioning on the reduced score above an adaptively well-chosen level, and ii) draws from the latter both for importance sampling of the true target rare events, as well as for proposing relevant (expensive) updates to the reduced score. An essential contribution consists in the adaptive choice of the level in i) and ii). The latter is calculated solely from the reduced score and its error bound, and is interpreted as the first non-achievable level as quantified by a given cost (in a pessimistic scenario) of importance sampling of the associated true target distribution. From a practical point of view, sampling the proposal sequence is performed by extending the framework of the popular Adaptive Multilevel Splitting (AMS) algorithm to the use of score function reduction. Numerical experiments evaluate the proposed importance sampling algorithm in terms of computational complexity versus squared error. In particular, we investigate the performance of the algorithm when simulating rare events related to the solution of a parametric PDE approximated by a reduced basis.

翻译：本文考虑采用蒙特卡洛方法对由罕见事件条件作用得到的目标概率分布进行采样的经典问题，该罕见事件由计算代价极高的实值评分函数的水平集定义。我们同时考虑一种场景：每次对真实评分函数进行新评估时，存在一种迭代构建简化评分序列的方法，并且这些简化评分具有逐点误差界的可验证保证。本文提出一种完全自适应的算法，该算法迭代执行以下步骤：i) 通过对适应选择的水平以上的简化评分进行条件作用，构建一系列建议分布；ii) 从上述分布中进行采样，既用于真实目标罕见事件的重要性采样，也用于提出（昂贵的）简化评分更新。核心贡献在于对步骤i)和ii)中水平的自适应选择。该水平仅通过简化评分及其误差界计算得出，可解释为在（悲观情景下）利用给定成本对相关真实目标分布进行重要性采样时，首个不可达的水平。从实践角度看，建议序列的采样通过将流行的自适应多层分裂（AMS）算法框架扩展到评分函数简化来实现。数值实验从计算复杂度与平方误差的角度评估了所提出的重要性采样算法。特别地，我们研究了该算法在模拟与参数化偏微分方程解相关的罕见事件时的性能，其中该方程的解通过简化基方法近似。