|
|
| Line 1: |
Line 1: |
| == Wiener Polynomial chaos ==
| | Alyson is the name people use to call me and I believe it seems fairly good when you say it. Invoicing is what I do. Some time in the past he selected to live in North Carolina and he doesn't plan on altering it. What me and my family members love is doing ballet but I've been taking on new things recently.<br><br>Feel free to surf to my page ... accurate psychic predictions; [http://black7.mireene.com/aqw/5741 Learn Alot more], |
| '''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 [[Volterra_series|theory]] of [[nonlinear]] functionals for stochastic systems. According to Cameron and Martin such an expansion converges in the <math>\mathcal{L}_2</math> sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems.
| |
| | |
| == Generalized polynomial chaos ==
| |
| Xiu generalized the result of Cameron-Martin to various continuous and discrete distributions using orthogonal polynomials from the so-called Askey-scheme and demonstrated <math>\mathcal{L}_2</math> 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.
| |
| | |
| == Arbitrary polynomial chaos ==
| |
| Recently chaos expansion received a generalization towards the arbitrary polynomial chaos expansion (aPC),<ref>[http://oladyshkin.net/publications/Oladyshkin_Nowak_RESS_2012.pdf Oladyshkin, S. and W. Nowak. Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety, Elsevier, V. 106, P. 179–190, 2012. DOI: 10.1016/j.ress.2012.05.002.]</ref> which is a so-called data-driven generalization of the PC. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. The aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Investigations indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques.
| |
| | |
| ==See also==
| |
| *[[Karhunen-Loève theorem]]
| |
| *[[Hilbert Space]]
| |
| *[[Proper Orthogonal Decomposition]]
| |
| | |
| ==References==
| |
| * {{cite journal |author=Wiener N. |title=The Homogeneous Chaos |journal=American Journal of Mathematics |volume=60 |issue=4 |pages=897–936 |year=1938 |month=October |doi=10.2307/2371268 |jstor=2371268 |publisher=American Journal of Mathematics, Vol. 60, No. 4}} (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
| |
| | |
| *Bin Wu, “A Statistically Optimal Macromodeling Framework with Application in Process Variation Analysis of MEMS Devices” IEEE 10th International New Circuits and Systems Conference (NEWCAS-12) June 2012
| |
| | |
| * K. Sepahvand, S. Marburg and H.-J. Hardtke, Uncertainty quantification in stochastic systems using polynomial chaos expansion, 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
| |
| * [http://oladyshkin.net/publications/Oladyshkin_Nowak_RESS_2012.pdf Oladyshkin, S. and W. Nowak. Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety, Elsevier, V. 106, P. 179–190, 2012. DOI: 10.1016/j.ress.2012.05.002.]
| |
| | |
| {{Reflist}}
| |
| | |
| {{DEFAULTSORT:Polynomial Chaos}}
| |
| [[Category:Stochastic processes]]
| |
Alyson is the name people use to call me and I believe it seems fairly good when you say it. Invoicing is what I do. Some time in the past he selected to live in North Carolina and he doesn't plan on altering it. What me and my family members love is doing ballet but I've been taking on new things recently.
Feel free to surf to my page ... accurate psychic predictions; Learn Alot more,