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An Adaptive Gaussian Mixture Model Approach Based Framework for Solving Fokker-Planck Kolmogorov Equation Related to High Dimensional Dynamical Systems

[+] Author Affiliations
Arpan Mukherjee, Rahul Rai, Puneet Singla, Tarunraj Singh, Abani Patra

University at Buffalo-SUNY, Buffalo, NY

Paper No. DETC2016-60312, pp. V02BT03A058; 8 pages
doi:10.1115/DETC2016-60312
From:
  • ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 2B: 42nd Design Automation Conference
  • Charlotte, North Carolina, USA, August 21–24, 2016
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5011-4
  • Copyright © 2016 by ASME

abstract

Engineering systems are often modeled as a large dimensional random process with additive noise. The analysis of such system involves a solution to simultaneous system of Stochastic Differential Equations (SDE). The exact solution to the SDE is given by the evolution of the probability density function (pdf) of the state vector through the application of Stochastic Calculus. The Fokker-Planck-Kolmogorov Equation (FPKE) provides approximate solution to the SDE by giving the time evolution equation for the non-Gaussian pdf of the state vector. In this paper, we outline a computational framework that combines linearization, clustering technique and the Adaptive Gaussian Mixture Model (AGMM) methodology for solving the Fokker-Planck-Kolmogorov Equation (FPKE) related to a high dimensional system. The linearization and clustering technique facilitate easier decomposition of the overall high dimensional FPKE system into a finite number of much lower dimension FPKE systems. The decomposition enables the solution method to be faster. Numerical simulations test the efficacy of our developed framework.

Copyright © 2016 by ASME

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