随机动力系统数值解谱统计特性评定误差的研究

Yaroslav Beskrovnii, Oleksii Larin
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引用次数: 0

摘要

本文提出了一种计算具有随机扰动的离散动力系统的数值模拟算法,并对数值解的质量进行了分析。为此,提出了一种具有随机右侧的二阶微分方程数值解的算法,并以程序形式实现了该算法。下一步是通过改变数值积分的参数进行一系列计算研究,随后评估它们对模拟误差和精度的影响。采用Welch周期图法估计谱密度。为了检查模拟的质量和评估解的准确性,建议将数值积分和随后的数字处理结果与已知的线性问题的解析解进行比较,由方程给出。作为工作的结果,位移的分散相对于来自不同数量的块的信号的长度进行了比较分析,其中的信号被划分为韦尔奇方法;不同信号长度下误差的置信区间和一定信号长度下不同块数误差的置信区间。不同块数的方差比较表明,在信号长度为30 s和从90 s开始,方差值有轻微的分散,误差在±5%以内。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Investigation of error in evaluation of spectral statistical characteristics of numerical solution of stochastic dynamic system
In this paper, an algorithm for numerical simulations is developed for calculating a discrete dynamic system with a stochastic perturbation and an analysis of the quality of numerical solutions is carried out. For this, an algorithm for the numerical solution of a second-order differential equation with a stochastic right-hand side was developed and this algorithm was implemented as a program. The next step was to carry out a set of computational studies by varying the parameters of numerical integration with the subsequent assessment of their impact on the error and accuracy of simulations. To estimate the spectral density, the Welch periodogram method was used. To check the quality of simulations and assess the accuracy of solutions, it is proposed to compare the results of numerical integration and subsequent digital processing with analytical solutions that are known for the linear problem, given by the equation. As a result of the work, a comparative analysis of the dispersion of displacements relative to the lengths of signals from a different number of blocks was carried out, into which the signal is divided for the Welch method; the confidence interval of the error at different signal lengths and the confidence interval of the error with a different number of blocks at a certain signal length. Comparison of the variance with a different number of blocks showed that with a signal length of 30 s and from 90 s, there is a slight scatter of the variance values within an error of ± 5%.
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