This paper considers the classical problem of sampling with Monte Carlo methods a target rare event distribution of the form $\eta^\star_{l_{max}} \propto \mathbb{1}_{S^\star> l_{max}} d\pi$, with $\pi$ a reference probability distribution, $S^\star$ a real-valued function that is very expensive to compute and $l_{max} \in \mathbb{R}$ some level of interest. We assume we can iteratively build a sequence of reduced models $\{S^{(k)}\}_k$ of the scores, which are increasingly refined approximations of $S^\star$ as new score values are computed; these reduced models being moreover certified with error bounds. This work proposes a fully adaptive algorithm to iteratively build a sequence of proposal distributions with increasing target levels $l^{(k)}$ of the form $\propto \mathbb{1}_{S^{(k)} > l^{(k)}} d\pi$ and draw from them, the objective being importance sampling of the target rare events, as well as proposing relevant updates to the reduced score. An essential contribution consists in {adapting} the target level to the reduced score: the latter is defined as the first non-achievable level $l^{(k)}$ corresponding to a cost, in a pessimistic scenario, for importance sampling matching an acceptable budget; it is calculated solely from the reduced score function and its error bound. From a practical point of view, sampling the proposal sequence is performed by extending the framework of the popular adaptive multilevel splitting algorithm to the use of reduced score approximations. 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, which is approximated by a reduced basis.

翻译：本文研究用蒙特卡洛方法对形如$\eta^\star_{l_{max}} \propto \mathbb{1}_{S^\star> l_{max}} d\pi$的目标稀有事件分布进行采样的经典问题，其中$\pi$为参考概率分布，$S^\star$为计算代价极高的实值函数，$l_{max} \in \mathbb{R}$为感兴趣阈值。我们假设可迭代构建分数函数的简化模型序列$\{S^{(k)}\}_k$，这些模型通过新分数值的计算逐步精化逼近$S^\star$，且此类简化模型具备带误差界的认证。本文提出一种全自适应算法，通过迭代构建形如$\propto \mathbb{1}_{S^{(k)} > l^{(k)}} d\pi$的递增目标水平$l^{(k)}$的提议分布序列并从中采样，其目标在于实现目标稀有事件的重要性采样，同时为简化分数的相关更新提供建议。核心贡献在于将目标水平与简化分数进行自适应匹配：该水平被定义为在悲观场景下，对应重要性采样成本不超过可接受预算时的首个不可达水平$l^{(k)}$，其计算仅依赖于简化分数函数及其误差界。从实践角度而言，提议序列的采样通过将流行的自适应多层次分割算法框架扩展至简化分数近似得以实现。数值实验从计算复杂度与平方误差的权衡角度评估了所提重要性采样算法，特别研究了该算法在模拟参数化偏微分方程解相关稀有事件时的性能表现，其中PDE解由简化基进行逼近。