This work combines multilevel Monte Carlo (MLMC) with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean--Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in (Ben Rached et al., 2023) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in (Ben Rached et al., 2023), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single-level DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$ while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$.

翻译：本文将多级蒙特卡洛（MLMC）与重要性采样相结合，估计与一类广泛McKean-Vlasov随机微分方程解的Lipschitz可观测量的期望相关的罕见事件量。我们将(Ben Rached等，2023)在此背景下引入的双环蒙特卡洛（DLMC）估计器推广至多级设定。我们提出了新型多级DLMC估计器，并进行了全面的成本误差分析，得出新的改进复杂度结果。关键在于，我们设计了反方差采样器来估计级间差异，确保多级DLMC估计器相比单级DLMC估计器具有更低的计算复杂度。为应对罕见事件，我们将通过(Ben Rached等，2023)中最优控制随机方法获得的重要性采样方案应用于多级DLMC估计器的所有层级。结合重要性采样与多级DLMC将计算复杂度降低一个量级，并将关联常数大幅缩减至低于无重要性采样的单级DLMC估计器。我们通过统计物理中具有Lipschitz可观测量的Kuramoto模型验证了所提多级DLMC估计器的有效性，确认复杂度从单级DLMC估计器的$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$降至$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$，同时能实现达到预设相对误差容限$\mathrm{TOL}_{\mathrm{r}}$的罕见事件量可行估计。