Systematic matrix formulation for efficient computational path integration

Henrik T. Sykora, Rachel Kuske, Daniil Yurchenko

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Abstract

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.
Original languageEnglish
Article number106896
JournalComputers and Structures
Volume273
Early online date7 Sep 2022
DOIs
Publication statusE-pub ahead of print - 7 Sep 2022

Keywords

  • Chapman Kolmogorov Equation
  • Convergence
  • Numerical Method
  • Path Integration Method
  • Probability Density Function
  • Random Dynamical System
  • Stochastic Differential Equations

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Modelling and Simulation
  • Materials Science(all)
  • Mechanical Engineering
  • Computer Science Applications

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