Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907, USA.
Department of Mathematics and Department of Physics, Duke University, Durham, North Carolina 27708, USA.
Chaos. 2023 Feb;33(2):023113. doi: 10.1063/5.0129127.
A new efficient ensemble prediction strategy is developed for a multiscale turbulent model framework with emphasis on the nonlinear interactions between large and small-scale variables. The high computational cost in running large ensemble simulations of high-dimensional equations is effectively avoided by adopting a random batch decomposition of the wide spectrum of the fluctuation states, which is a characteristic feature of the multiscale turbulent systems. The time update of each ensemble sample is then only subject to a small portion of the small-scale fluctuation modes in one batch, while the true model dynamics with multiscale coupling is respected by frequent random resampling of the batches at each time updating step. We investigate both theoretical and numerical properties of the proposed method. First, the convergence of statistical errors in the random batch model approximation is shown rigorously independent of the sample size and full dimension of the system. Next, the forecast skill of the computational algorithm is tested on two representative models of turbulent flows exhibiting many key statistical phenomena with a direct link to realistic turbulent systems. The random batch method displays robust performance in capturing a series of crucial statistical features with general interests, including highly non-Gaussian fat-tailed probability distributions and intermittent bursts of instability, while requires a much lower computational cost than the direct ensemble approach. The efficient random batch method also facilitates the development of new strategies in uncertainty quantification and data assimilation for a wide variety of general complex turbulent systems in science and engineering.
提出了一种新的高效集成预测策略，用于多尺度湍流模型框架，重点关注大尺度和小尺度变量之间的非线性相互作用。通过采用波动状态的宽频谱的随机批量分解，有效地避免了高维方程的大集合模拟的高计算成本，这是多尺度湍流系统的一个特征。然后，通过在每个时间更新步骤中频繁随机重新采样批处理，仅对一个批处理中的一小部分小尺度波动模式进行每个集合样本的时间更新，同时通过频繁随机重新采样批处理来尊重具有多尺度耦合的真实模型动力学。我们研究了所提出方法的理论和数值特性。首先，严格证明了随机批量模型逼近的统计误差的收敛性与样本大小和系统的全维数无关。接下来，在两个具有直接关联到实际湍流系统的许多关键统计现象的湍流流动的代表性模型上测试了计算算法的预测技巧。随机批量方法在捕捉具有普遍兴趣的一系列关键统计特征方面表现出稳健的性能，包括高度非高斯的长尾概率分布和不稳定的间歇性爆发，同时比直接集合方法要求低得多的计算成本。高效的随机批量方法还为科学和工程中各种复杂的一般湍流系统的不确定性量化和数据同化的发展提供了新的策略。
