Systematic matrix formulation for efficient computational path integration
Systematic matrix formulation for efficient computational path integration
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.
Chapman Kolmogorov Equation, Convergence, Numerical Method, Path Integration Method, Probability Density Function, Random Dynamical System, Stochastic Differential Equations
Sykora, Henrik
e89d4c51-f8bc-4258-a9cc-38f249270757
Kuske, Rachel
eb2504e2-25b3-4838-8c59-8fe8eaecf443
Yurchenko, Daniil
51a2896b-281e-4977-bb72-5f96e891fbf8
1 December 2022
Sykora, Henrik
e89d4c51-f8bc-4258-a9cc-38f249270757
Kuske, Rachel
eb2504e2-25b3-4838-8c59-8fe8eaecf443
Yurchenko, Daniil
51a2896b-281e-4977-bb72-5f96e891fbf8
Sykora, Henrik, Kuske, Rachel and Yurchenko, Daniil
(2022)
Systematic matrix formulation for efficient computational path integration.
Computers & Structures, 273, [106896].
(doi:10.1016/j.compstruc.2022.106896).
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.
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Accepted/In Press date: 24 August 2022
e-pub ahead of print date: 7 September 2022
Published date: 1 December 2022
Additional Information:
Funding Information:
The authors gratefully acknowledge partial funding for this work from NSF-CMMI 2009270 and EPSRC EP/V034391/1.
Publisher Copyright:
© 2022 The Authors
Keywords:
Chapman Kolmogorov Equation, Convergence, Numerical Method, Path Integration Method, Probability Density Function, Random Dynamical System, Stochastic Differential Equations
Identifiers
Local EPrints ID: 470624
URI: http://eprints.soton.ac.uk/id/eprint/470624
ISSN: 0045-7949
PURE UUID: 97720813-cbd2-485d-89e8-b492255f2f8d
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Date deposited: 14 Oct 2022 16:51
Last modified: 06 Jun 2024 02:12
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Contributors
Author:
Henrik Sykora
Author:
Rachel Kuske
Author:
Daniil Yurchenko
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