A homotopic approach to domain determination and solution refinement for the stationary Fokker-Planck equation

Probabilistic Engineering Mechanics

Volumn 24, Issue 3, 2009, Pages 265-277

  Kumar, Mrinal ^a   Chakravorty, Suman ^a   Junkins, John L ^a  

^a Department of Electrical and Computer Engineering   (United States)

Author keywords

Galerkin projection; Hilbert space approximation; Homotopic recursion; Stationary Fokker Planck equation

Indexed keywords

DYNAMICAL SYSTEMS; GALERKIN METHODS; HILBERT SPACES; NONLINEAR SYSTEMS; PROBABILITY DENSITY FUNCTION; RECONNAISSANCE AIRCRAFT; RECURSIVE FUNCTIONS; TIMING JITTER;

APPROXIMATION ACCURACIES; DEGREES OF FREEDOMS; ERROR PROJECTIONS; GALERKIN; GALERKIN PROJECTION; HIGHER DIMENSIONS; HIGHER-DIMENSIONAL PROBLEMS; HOMOTOPIC RECURSION; HOMOTOPY; ITERATIVE APPROACHES; MESHLESS; NUMERICAL IMPLEMENTATIONS; PARTITION OF UNITY FINITE ELEMENT METHODS; PROBABILITY DENSITIES; STATIONARY DISTRIBUTIONS; STATIONARY FOKKER-PLANCK EQUATION;

FOKKER PLANCK EQUATION;

EID: 63449140437     PISSN: 02668920     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.probengmech.2008.07.006     Document Type: Article

References (22)

1. Risken H. The Fokker Planck equation: Methods of solution and applications. Springer series in synergetics (1989), Springer-Verlag
2. Kumar M, Singla P, Chakravorty S, Junkins JL. The partition of unity finite element approach to the staitonary Fokker-Planck equation. Journal of Guidance and Control Design, also Paper 2006-6285 at the 2006 AIAA/AAS astrodynamics specialist conf. [submitted for publication]
3. Fuller A.T. Analysis of nonlinear stochastic systems by means of the Fokker-Planck equation. Int J Control 9 6 (1969) 603-655
4. Johnson E.A., Wojtkiewicz S.F., Bergman L.A., and Spencer B.F. Observations with regard to massively parallel computation for Monte Carlo simulation of stochastic dynamical systems. Int J Non-linear Mech 32 4 (1997) 721-734
5. Harnpornchai N., Pradlwarter H.J., and Schnëller G.I. Stochastic analysis of dynamical systems by phase-space-controlled Monte Carlo simulation. Comput Methods Appl Mech Engr 168 (1999) 273-283
6. Roberts J.B., and Spanos P.D. Random vibration and statistical linearization (2003), Dover Publications
7. Wu W.F., and Lin Y.K. Cumulant-neglect closure for nonlinear oscillators under parametric and external excitations. Int J Non-linear Mech 19 4 (1984) 349-362
8. Pradlwarter HJ, Vasta M. Numerical solution of the Fokker-Planck equation via Gaussian superposition representation. In: Shairaishi N, Shinozuka M, Wen YK, editors. Structural safety and reliability: Proc. ICOSSAR'97 - 7th int. conf. on structural safety and reliability, vol. 2, 1997, p. 917-23
9. Park RS, Scheers D. Nonlinear mapping of Gaussian state uncertainties: Theory and applications to spacecraft control and navigation. In: AAS/AIAA astrodynamics specialist conference, AAS paper no. 05-404
10. Pradlwarter H.J. Nonlinear stochastic response distributions by local statistical linearization. Int J Non-linear Mech 36 7 (2001) 1135-1151
11. Muscolino G., Ricciardi G., and Vasta M. Stationary and non stationary probability density function for non-linear oscillators. Int J Non-linear Mech 32 6 (1997) 1051-1064
12. Paola M.Di., and Sofi A. Approximate solution of the Fokker-Planck-Kolmogorov equation. Probabilist Eng Mech 17 (2002) 369-384
13. Wojtkiewicz SF, Bergman LA, Spencer BF Jr. High fidelity numerical solutions of the Fokker-Planck equation. In: Bazzani A, Ellison J, Mais H, Turchetti G, editors. Proc. of ICOSSAR'97: The 7th international conference on structural safety and reliability, 1997
14. Johnson EA, Wojtkiewicz SF, Bergman LA, SpencerJr BF. Finite element and finite difference solutions to the transient Fokker-Planck equation. In: Bazzani A, Ellison J, Mais H Turchetti G, editors. Proc. of a workshop: Nonlinear and stochastic beam dynamics in accelerators - a challenge to theoretical and numerical physics, 1997, p. 290-306
15. Masud A, Bergman LA. Application of multi-scale finite element methods to the solution of the Fokker-Planck equation. Comput Methods Appl Mech Engr [in press]
16. Langley R.S. A finite element method for the statistics of random nonlinear vibration. J Sound Vib 101 1 (1985) 41-54
17. Kumar M, Singla P, Junkins JL, Chakravorty S. A multi-resolution meshless approach to steady state uncertainty determination in nonlinear dynamical systems. In: 38th IEEE southeastern symp. on sys. theory. 2006
18. Chakravorty S, Kumar M, Singla P. A quasi-Gaussian Kalman filter. In: American control conference. 2006
19. Singla P. Multi-resolution methods for high fidelity modeling and control allocation in large-scale dynamical systems. Ph.D. dissertation. College Station (USA) : Department of Aerospace Engineering. Texas A&M University, 2006
20. Babuška I., and Melenk J. The partition of unity finite element method. Int J Num Meth 40 (1997) 727-758
21. Atluri S.N., and Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in Computational Mechanics. Comput Mech 22 2 (1998) 117-127
22. Wojtkiewicz SF, Bergman LA, Spencer BF Jr. Numerical solution of some three state random vibration problems. In: Bergman LA, Spencer Jr BF, editors. Vibration of nonlinear, random and time-varying system. 1995