论文标题
部分可观测时空混沌系统的无模型预测
An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox
论文作者
论文摘要
为了进行双重迭代的随机积分和相应的莱维区W.R.T.的近似和模拟。一个多维的维纳过程，我们根据傅立叶系列方法回顾了四种算法。特别是，由于Mrongowius和Rössler引起的新提出的算法，尤其是由于Wiktorsson引起的非常有效的算法。为了将最近的进步融入上下文，我们在统一框架中分析了四种基于傅立叶的算法，以突出其派生中的差异和相似之处。理论属性的比较是通过数值模拟来补充的，该数值模拟揭示了每种算法的收敛顺序。此外，给出了选择最佳算法和参数的具体指令，以模拟用于随机（部分）微分方程的解决方案。此外，我们为有效实施了所考虑的算法提供建议，并将这些见解纳入开源工具箱中，该工具箱可免费用于Julia和Matlab编程语言。通过将其与某些现有实现进行比较来分析该工具箱的性能，我们观察到大幅加速。
For the approximation and simulation of twofold iterated stochastic integrals and the corresponding Lévy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and Rössler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and MATLAB programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up.
