微-宏加速方法的分析与最小相对熵矩匹配
Analysis of a micro-macro acceleration method with minimum relative entropy moment matching
AI总结：
本文分析了微-宏加速方法在处理具有时间尺度分离的随机微分方程蒙特卡罗模拟中的收敛性，通过最小化相对熵来优化宏观状态变量的扰动，研究了方法的局部误差和数值稳定性。
摘要：
我们分析了用于随机微分方程蒙特卡罗模拟的微-宏加速方法的收敛性，该方法在个体轨迹的快速演化和目标宏观函数的慢速演化之间存在时间尺度分离。我们考虑了一类方法，该方法通过短时间路径模拟和宏观状态变量的向前外推相结合，然后构造新的微观状态，使其与外推变量一致并最小化由外推引起的扰动。在本文中，我们研究了一种特定方法，其中扰动通过相对熵进行最小化。我们讨论了相对熵作为度量标准的理论和实际价值，并严格研究了该方法的局部误差和数值稳定性，作为外推时间步长和宏观状态变量数量的函数。利用这些结果，我们讨论了当外推时间步长趋于零且宏观状态变量数量趋于无穷时，向完整微观动态收敛的情况。
英文摘要：
We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest. We consider a class of methods, presented in [Debrabant, K., Samaey, G., Zieliński, P. A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short bursts of path simulations, combined with the extrapolation of a few macroscopic state variables forward in time. After extrapolation, a new microscopic state is then constructed, consistent with the extrapolated variable and minimising the perturbation caused by the extrapolation. In the present paper, we study a specific method in which this perturbation is minimised in a relative entropy sense. We discuss why relative entropy is a useful metric, both from a theoretical and practical point of view, and rigorously study local errors and numerical stability of the resulting method as a function of the extrapolation time step and the number of macroscopic state variables. Using these results, we discuss convergence to the full microscopic dynamics, in the limit when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.
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