This work addresses the estimation of rare-event quantities expressed as expectations of smooth observables of solutions to a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs). Building on the double loop Monte Carlo (DLMC) method with stochastic optimal control-based importance sampling (IS) introduced by Ben Rached et al. (2024a), this work extends this framework to the multi-index Monte Carlo (MIMC) setting. The resulting multi-index DLMC estimator mitigates the explosion of the coefficient of variation for rare event quantities. Moreover, it exploits the sampling efficiency of MIMC by leveraging the propagation of chaos to ensure mixed-difference variances vanish in the mean-field limit. The complexity analysis relies on assumptions on mixed-difference bias and variance decay, similar to standard MIMC assumptions. Although not rigorously proved, this work presents strong numerical evidence in support of these assumptions. The primary contribution of this work is the novel numerical integration of the MIMC method with IS for MV-SDEs. This approach reduces the computational complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$, enabling an accurate estimation of rare-event quantities within a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Numerical experiments on the Kuramoto model from statistical physics demonstrate computational savings of several orders of magnitude for the multi-index DLMC estimator with IS, compared with the standard Monte Carlo (MC) method.

翻译：本研究致力于估计一类广泛的McKean--Vlasov随机微分方程（MV-SDEs）解的光滑观测量的期望所表示的罕见事件量。基于Ben Rached等人（2024a）提出的结合随机最优控制重要性采样（IS）的双层蒙特卡洛（DLMC）方法，本研究将该框架扩展至多指标蒙特卡洛（MIMC）设置。由此产生的多指标DLMC估计量缓解了罕见事件量变异系数的爆炸性增长。此外，该方法通过利用混沌传播确保混合差分方差在平均场极限下消失，从而充分发挥MIMC的采样效率。复杂度分析依赖于关于混合差分偏差和方差衰减的假设，类似于标准MIMC假设。尽管未严格证明，本研究提供了支持这些假设的有力数值证据。本研究的主要贡献在于首次将MIMC方法与IS相结合，用于MV-SDEs的数值积分。该方法将计算复杂度从DLMC估计量的$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$降低至$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$，从而能够在规定的相对误差容限$\mathrm{TOL}_{\mathrm{r}}$内精确估计罕见事件量。基于统计物理学中Kuramoto模型的数值实验表明，与标准蒙特卡洛（MC）方法相比，采用IS的多指标DLMC估计量实现了数个数量级的计算节省。