Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro–macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages forward in time. To have explicit derivations, we elicit an amenable linear test equation containing multiple time scales. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. The analysis shows that, for this test model, the stability threshold on the extrapolation step is largely independent of the time-scale separation. In consequence, the micro–macro acceleration method increases the admissible time steps far beyond those for which a direct time discretization becomes unstable.

中文翻译：

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具有加性噪声的线性慢-快随机微分方程的微-宏加速方案研究

计算多尺度方法利用大的时间尺度分离来有效地模拟长时间间隔内的慢动态。对于随机系统，通常旨在解决最慢动态的统计数据。本文研究了一种微宏观加速方法的效率，该方法将随机路径模拟的短突发与空间平均值的向前外推及时相结合。为了有明确的推导，我们引出一个包含多个时间尺度的适合的线性测试方程。我们在高斯设置中进行推导并进行数值实验，其中只有均值和方差的演变很重要。分析表明，对于这个测试模型，外推步骤的稳定性阈值在很大程度上与时间尺度分离无关。结果，