This work introduces a novel approach that combines the multi-index Monte Carlo (MC) method with importance sampling (IS) to estimate rare event quantities expressed as an expectation of a smooth observable of solutions to a broad class of McKean-Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator, previously introduced in our works (Ben Rached et al., 2022a,b), to the multi-index setting. We formulate a new multi-index DLMC estimator and conduct a comprehensive cost-error analysis, leading to improved complexity results. To address rare events, an importance sampling scheme is applied using stochastic optimal control of the single level DLMC estimator. This combination of IS and multi-index DLMC not only reduces computational complexity by two orders but also significantly decreases the associated constant compared to vanilla MC. The effectiveness of the proposed multi-index DLMC estimator is demonstrated using the Kuramoto model from statistical physics. The results confirm a reduced complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single level DLMC estimator (Ben Rached et al., 2022a) to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$ for the considered example, while ensuring accurate estimation of rare event quantities within the prescribed relative error tolerance $\mathrm{TOL}_\mathrm{r}$.

翻译：本文提出了一种新方法，将多指标蒙特卡洛方法与重要性抽样相结合，用于估计一类广义Mckean-Vlasov随机微分方程解的平滑可观测量的期望所表达的稀有事件量。我们将先前工作中（Ben Rached等，2022a,b）引入的双层蒙特卡洛估计器推广到多指标框架。我们构建了新的多指标双层蒙特卡洛估计器，并对其进行了全面的成本-误差分析，从而获得了改进的复杂度结果。为处理稀有事件，采用基于随机最优控制的单层双层蒙特卡洛估计器实施重要性抽样方案。这种重要性抽样与多指标双层蒙特卡洛的结合不仅将计算复杂度降低两个数量级，而且相较于普通蒙特卡洛方法，显著减小了相关常数。通过统计物理学中的Kuramoto模型验证了所提出的多指标双层蒙特卡洛估计器的有效性。结果表明，对于所考虑的实例，复杂度从单层双层蒙特卡洛估计器（Ben Rached等，2022a）的$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$降低至$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$，同时在规定的相对误差容限$\mathrm{TOL}_\mathrm{r}$内确保了稀有事件量的精确估计。