An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems

Arpan Mukherjee, Rahul Rai, Puneet Singla, Tarunraj Singh, Abani Patra

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publication42nd Design Automation Conference
PublisherAmerican Society of Mechanical Engineers (ASME)
ISBN (Electronic)9780791850114
DOIs
StatePublished - Jan 1 2016
EventASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2016 - Charlotte, United States
Duration: Aug 21 2016Aug 24 2016

Publication series

NameProceedings of the ASME Design Engineering Technical Conference
Volume2B-2016

Other

OtherASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2016
CountryUnited States
CityCharlotte
Period8/21/168/24/16

Fingerprint

Fokker Planck equation
Kolmogorov Equation
Gaussian Mixture Model
Fokker-Planck Equation
Dynamical systems
High-dimensional
Dynamical system
Differential equations
Stochastic Equations
Linearization
Probability density function
Differential equation
Decomposition
Clustering
Additive noise
Decompose
Systems engineering
Random processes
Stochastic Calculus
Additive Noise

All Science Journal Classification (ASJC) codes

  • Mechanical Engineering
  • Computer Graphics and Computer-Aided Design
  • Computer Science Applications
  • Modeling and Simulation

Cite this

Mukherjee, A., Rai, R., Singla, P., Singh, T., & Patra, A. (2016). An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems. In 42nd Design Automation Conference (Proceedings of the ASME Design Engineering Technical Conference; Vol. 2B-2016). American Society of Mechanical Engineers (ASME). https://doi.org/10.1115/DETC2016-60312
Mukherjee, Arpan ; Rai, Rahul ; Singla, Puneet ; Singh, Tarunraj ; Patra, Abani. / An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems. 42nd Design Automation Conference. American Society of Mechanical Engineers (ASME), 2016. (Proceedings of the ASME Design Engineering Technical Conference).
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Mukherjee, A, Rai, R, Singla, P, Singh, T & Patra, A 2016, An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems. in 42nd Design Automation Conference. Proceedings of the ASME Design Engineering Technical Conference, vol. 2B-2016, American Society of Mechanical Engineers (ASME), ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2016, Charlotte, United States, 8/21/16. https://doi.org/10.1115/DETC2016-60312

An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems. / Mukherjee, Arpan; Rai, Rahul; Singla, Puneet; Singh, Tarunraj; Patra, Abani.

42nd Design Automation Conference. American Society of Mechanical Engineers (ASME), 2016. (Proceedings of the ASME Design Engineering Technical Conference; Vol. 2B-2016).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Mukherjee A, Rai R, Singla P, Singh T, Patra A. An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems. In 42nd Design Automation Conference. American Society of Mechanical Engineers (ASME). 2016. (Proceedings of the ASME Design Engineering Technical Conference). https://doi.org/10.1115/DETC2016-60312