论文标题
对弱相互作用扩散的经验度量的重要性抽样
Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions
论文作者
论文摘要
我们构建了一种重要的抽样方法，用于计算与稀有事件相关的弱相互作用扩散的统计数据。对于此类问题，标准蒙特卡洛方法的行为呈指数级别，系统中的颗粒数量很差。我们的方案基于在均值场（McKean-Vlasov）控制理论中产生的汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程的亚树种。我们确定这种方案在渐近上最佳的条件。在此过程中，我们建立了大偏差原理之间的连接，以实现弱相互作用扩散，平均场控制和HJB方程的经验度量。我们还提供了分析和数值的证据，表明凭借足够的HJB方程式，我们的方案在许多粒子极限上可能会消失小相对误差。
We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB Equation on Wasserstein Space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB Equation, our scheme can have vanishingly small relative error in the many particle limit.
