Polynomial chaos

Polynomial chaos (PC) , also called "Wiener Chaos expansion", is a non-sampling based method to determine evolution of uncertainty in dynamical system, when there is probabilistic uncertainty in the system parameters.

PC was first introduced by Wiener where Hermite polynomials were used to model stochastic processes with Gaussian random variables. It can be thought of as an extension of Volterra's theory of nonlinear functionals for stochastic systems. According to Cameron and Martin such an expansion converges in the sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems. Xiu generalized the result of Cameron-Martin to various continuous and discrete distributions using orthogonal polynomials from the so called Askey-scheme and demonstrated convergence in the corresponding Hilbert functional space. This is popularly known as the generalized polynomial chaos (gPC) framework. The gPC framework has been applied to applications including stochastic fluid dynamics, stochastic finite elements, solid mechanics, nonlinear estimation, and probabilistic robust control. It has been demonstrated that gPC based methods are computationally superior to Monte-Carlo based methods in a number of applications. However, the method has a notable limitation. For large numbers of random variables, polynomial chaos becomes very computationally expensive and Monte-Carlo methods are typically more feasible.

See also

• Karhunen-Loève theorem
• Hilbert Space
• Proper Orthogonal Decomposition

References

• Wiener N. (October 1938). "The Homogeneous Chaos". American Journal of Mathematics (American Journal of Mathematics, Vol. 60, No. 4) 60 (4): 897–936. doi:10.2307/2371268. JSTOR 2371268.  (original paper)

• D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach Princeton University Press, 2010. ISBN 978-0-691-14212-8

• Ghanem, R., and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer Verlag, 1991. (reissued by Dover Publications, 2004.)

• Bin Wu, Jianwen Zhu, Farid N. Najm. "A Non-parametric Approach for Dynamic Range Estimation of Nonlinear Systems". In Proceedings of Design Automation Conference(841-844) 2005

• Bin Wu, Jianwen Zhu, Farid N. Najm "Dynamic Range Estimation". IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 25 Issue:9 (1618-1636) 2006

• Uncertainty quantification in stochastic systems using polynomial chaos expansion– K. Sepahvand, S. Marburg and H.-J. Hardtke, International Journal of Applied Mechanics, vol. 2, No. 2,p. 305-353, 2010.

• Nonlinear Estimation of Hypersonic State Trajectories in Bayesian Framework with Polynomial Chaos – P. Dutta, R. Bhattacharya, Journal of Guidance, Control, and Dynamics, vol.33 no.6 (1765–1778).

• Optimal Trajectory Generation with Probabilistic System Uncertainty Using Polynomial Chaos – J. Fisher, R. Bhattacharya, Journal of Dynamic Systems, Measurement and Control, volume 133, Issue 1.

• Linear Quadratic Regulation of Systems with Stochastic Parameter Uncertainties – J. Fisher, R. Bhattacharya, Automatica, 2009.

• E. Blanchard, A. Sandu, and C. Sandu: "Polynomial Chaos Based Parameter Estimation Methods for Vehicle Systems". Journal of Multi-body dynamics, in print, 2009.

• H. Cheng and A. Sandu: "Efficient Uncertainty Quantification with the Polynomial Chaos Method for Stiff Systems". Computers and Mathematics with Applications, VOl. 79, Issue 11, p. 3278-3295, 2009.

• Peccati, G. and Taqqu, M.S., 2011, Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation. Springer Verlag.

• Stochastic Processes and Orthogonal Polynomials Series: Lecture Notes in Statistics, Vol. 146 by Schoutens, Wim, 2000, XIII, 184 p., Softcover ISBN 978-0-387-95015-0

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