与McKean--Vlasov方程相关的罕见事件的多层重要性采样
Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation
AI总结：
针对McKean-Vlasov方程解的罕见事件估计问题，提出多层双循环蒙特卡洛估计器，结合反演采样器与基于随机最优控制的重要性采样方案，将计算复杂度降低一个数量级。
摘要：
本研究将多层蒙特卡洛（MLMC）与重要性采样相结合，用于估计罕见事件量，这些量可以表示为一大类McKean--Vlasov随机微分方程解的Lipschitz可观测量的期望。我们将（Ben Rached et al., 2023）在此背景下引入的双循环蒙特卡洛（DLMC）估计器扩展到多层设置。我们提出了一种新颖的多层DLMC估计器，并进行了全面的成本-误差分析，得出了新的且改进的复杂性结果。关键的是，我们设计了一种反演采样器来估计层级差异，与单层DLMC估计器相比，保证了多层DLMC估计器计算复杂度的降低。为了处理罕见事件，我们在多层DLMC估计器的所有层级上应用了通过随机最优控制获得的重要性采样方案（Ben Rached et al., 2023）。将重要性采样与多层DLMC相结合，计算复杂度降低了一个数量级，并且与不带重要性采样的单层DLMC估计器相比，相关常数大幅减小。我们在统计物理学中具有Lipschitz可观测量的Kuramoto模型上说明了所提出的多层DLMC估计器的有效性，证实了复杂度从单层DLMC估计器的$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$降低到$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$，同时提供了高达规定相对误差容限$\mathrm{TOL}_{\mathrm{r}}$的罕见事件量的可行估计。
英文摘要：
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 $\mathrm{TOL}_{\mathrm{r}}$.
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