Modified Optimal Prediction and its Application to a Particle-Method Problem

The paper is concerned with system reduction by statistical methods and, in particular, by the optimal prediction method introduced in (Chorin, A.J., Hald, O.H., Kupferman, R., Optimal prediction with memory, Phys. D 166:239–257, 2002). The optimal prediction method deals with problems that possess large and small scales and uses the conditional expectation to model the influence of the small scales on the large ones.In the current paper, we develop a different variant of the optimal prediction method as well as introduce and compare several approximations of this method. We apply the original and modified optimal prediction methods to a system of ODEs obtained from a particle method discretization of a hyperbolic PDE and demonstrate their performance in a number of numerical experiments.

本文关注通过统计方法进行系统降阶，特别是通过（乔林，A.J.，哈尔德，O.H.，库普弗曼，R.，《具有记忆的最优预测》，《物理D》166：239 - 257，2002年）中引入的最优预测方法。最优预测方法处理具有大尺度和小尺度的问题，并使用条件期望来模拟小尺度对大尺度的影响。在本文中，我们开发了最优预测方法的一种不同变体，并介绍和比较了该方法的几种近似。我们将原始的和改进的最优预测方法应用于从双曲型偏微分方程的粒子方法离散化得到的常微分方程组，并在一些数值实验中展示了它们的性能。