Systematic matrix formulation for efficient computational path integration
Systematic matrix formulation for efficient computational path integration
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DOI：
10.1016/j.compstruc.2022.106896
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发表时间：
2022-12
期刊：
影响因子：
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通讯作者：
Henrik T. Sykora;R. Kuske;D. Yurchenko
中科院分区：
文献类型：
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作者：
Henrik T. Sykora;R. Kuske;D. Yurchenko
在这项工作中，我们介绍了一种新的方法处理数值路径积分法，用于计算随机动力系统的响应概率密度函数。将相应的Chapman-Kolmogorov方程转化为矩阵乘法，大大加快了求解速度。用系统的公式将Chapman-Kolmogorov方程的数值解分成三个独立的部分：插值概率密度函数，近似过程的过渡概率密度函数和计算Chapman-Kolmogorov方程中的积分。我们通过对一、二、三、四维问题的数值实验，对误差和效率进行了全面的分析。通过将路径积分法所得结果与解析解及路径积分法的以往公式进行比较，证明了该公式提供准确结果的优越能力。指出了潜在的瓶颈，并讨论了如何解决这些瓶颈。
In this work we introduce a novel methodological treatment of the numerical path integration method, used for computing the response probability density function of stochastic dynamical systems. The method is greatly accelerated by transforming the corresponding Chapman-Kolmogorov equation to a matrix multiplication. With a systematic formulation we split the numerical solution of the Chapman-Kolmogorov equation into three separate parts: we interpolate the probability density function, we approximate the transitional probability density function of the process and evaluate the integral in the Chapman-Kolmogorov equation. We provide a thorough error and efficiency analysis through numerical experiments on a one, two, three and four dimensional problem. By comparing the results obtained through the Path Integration method with analytical solutions and with previous formulations of the path integration method, we demonstrate the superior ability of this formulation to provide accurate results. Potential bottlenecks are identified and a discussion is provided on how to address them.