### 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 language | English (US) |
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Title of host publication | 42nd Design Automation Conference |

Publisher | American Society of Mechanical Engineers (ASME) |

ISBN (Electronic) | 9780791850114 |

DOIs | |

State | Published - Jan 1 2016 |

Event | ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2016 - Charlotte, United States Duration: Aug 21 2016 → Aug 24 2016 |

### Publication series

Name | Proceedings of the ASME Design Engineering Technical Conference |
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Volume | 2B-2016 |

### Other

Other | ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2016 |
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Country | United States |

City | Charlotte |

Period | 8/21/16 → 8/24/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - An adaptive Gaussian mixture model approach based framework for solving fokker-planck kolmogorov equation related to high dimensional dynamical systems

AU - Mukherjee, Arpan

AU - Rai, Rahul

AU - Singla, Puneet

AU - Singh, Tarunraj

AU - Patra, Abani

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85007621929&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85007621929&partnerID=8YFLogxK

U2 - 10.1115/DETC2016-60312

DO - 10.1115/DETC2016-60312

M3 - Conference contribution

AN - SCOPUS:85007621929

T3 - Proceedings of the ASME Design Engineering Technical Conference

BT - 42nd Design Automation Conference

PB - American Society of Mechanical Engineers (ASME)

ER -