We analyze the convergence and stability of a micro–macro acceleration algorithm for Monte Carlo simulations of linear stiff stochastic differential equations with a time-scale separation between the fast evolution of the individual stochastic realizations and some slow macroscopic state variables of the process. The micro–macro acceleration method performs a short simulation of a large ensemble of individual fast paths, before extrapolating the macroscopic state variables of interest over a larger time step. After extrapolation, the method constructs a new probability distribution that is consistent with the extrapolated macroscopic state variables, while minimizing Kullback–Leibler divergence with respect to the distribution available at the end of the Monte Carlo simulation. In the current work, we study the convergence and stability of this method on linear stochastic differential equations with additive noise, when only extrapolating the mean of the slow component. For this case, we prove convergence to the microscopic dynamics when the initial distribution is Gaussian and present a stability result for non-Gaussian initial laws.

我们分析了用于线性刚性随机微分方程的蒙特卡罗模拟的微宏加速算法的收敛性和稳定性，该过程在各个随机实现的快速演化与过程的一些慢速宏观状态变量之间存在时间尺度上的分隔。微观-宏加速方法对单个快速路径的大集合进行简短的仿真，然后在较大的时间步长上推断出感兴趣的宏观状态变量。外推后，该方法构建与外推的宏观状态变量一致的新概率分布，同时相对于蒙特卡洛模拟结束时可用的分布，使Kullback-Leibler散度最小。在目前的工作中，当仅外推慢分量的均值时，我们研究了该方法对具有加性噪声的线性随机微分方程的收敛性和稳定性。对于这种情况，当初始分布为高斯分布时，我们证明了微观动力学的收敛性，并给出了非高斯初始定律的稳定性结果。