微-宏观加速方法在分尺度随机微分方程中的效率
Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations
AI总结：
本文研究了针对具有时间尺度分离的刚性随机微分方程的微-宏观加速方法的精度和效率，通过数值实验分析了不同 extrapolation 时间步长和所选宏观状态变量对方法性能的影响，并比较了不同层次的宏观状态变量的效果。
摘要：
我们通过多个数值例子讨论了针对具有快的微观动态和慢的宏观状态变量演化的刚性随机微分方程（SDEs）的微-宏观加速方法的精度和效率。该算法将刚性SDE的短时间模拟与宏观状态变量的长间隔外推相结合。外推后，我们通过匹配过程获得重构的微观状态：我们计算与外推状态变量一致且在外推前微观分布最小改变的概率分布。在本工作中，我们数值研究了微-宏观加速方法的精度和效率，作为外推时间步长和所选宏观状态变量函数。此外，我们比较了不同层次的宏观状态变量的效果。我们证明该方法可以比内部微观积分器显著更大的时间步长，同时比近似宏观模型更准确。
英文摘要：
We discuss through multiple numerical examples the accuracy and efficiency of a micro-macro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of some slow macroscopic state variables. The algorithm interleaves a short simulation of the stiff SDE with extrapolation of the macroscopic state variables over a longer time interval. After extrapolation, we obtain the reconstructed microscopic state via a matching procedure: we compute the probability distribution that is consistent with the extrapolated state variables, while minimally altering the microscopic distribution that was available just before the extrapolation. In this work, we numerically study the accuracy and efficiency of micro-macro acceleration as a function of the extrapolation time step and as a function of the chosen macroscopic state variables. Additionally, we compare the effect of different hierarchies of macroscopic state variables. We illustrate that the method can take significantly larger time steps than the inner microscopic integrator, while simultaneously being more accurate than approximate macroscopic models.
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