# Polynomial chaos

**Polynomial chaos (PC)**, also called **Wiener chaos expansion**, is a non-sampling-based method to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. PC was first introduced by Norbert Wiener using Hermite polynomials 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.

## Generalized polynomial chaosEdit

Xiu (in his PhD under Karniadakis at Brown University) 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, the evaluation of finite word-length effects in non-linear fixed-point digital systems 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^{[citation needed]}. 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^{[citation needed]}.

## Arbitrary polynomial chaosEdit

Recently chaos expansion received a generalization towards the arbitrary polynomial chaos expansion (aPC),^{[1]} 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. Yet these techniques are in progress but the impact of them on CFD models is quite impressionable.

## Polynomial chaos & incomplete statistical informationEdit

In many practical situations, only incomplete and inaccurate statistical knowledge on uncertain input parameters are available. Fortunately, to construct a finite-order expansion, only some partial information on the probability measure is required that can be simply represented by a finite number of statistical moments. Any order of expansion is only justified if accompanied by reliable statistical information on input data. Thus, incomplete statistical information limits the utility of high-order polynomial chaos expansions^{[2]}.

## Polynomial chaos & non-linear predictionEdit

Polynomial chaos can be utilized in the prediction of non-linear functionals of Gaussian stationary increment processes conditioned on their past realizations ^{[3]}. Specifically, such prediction is obtained by deriving the chaos expansion of the functional with respect to a special basis for the Gaussian Hilbert space generated by the process that with the property that each basis element is either measurable or independent with respect to the given samples. For example, this approach leads to an easy prediction formula for the Fractional Brownian motion.

## Software toolsEdit

## See alsoEdit

## ReferencesEdit

- 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) - Cameron, R. H.; Martin, W. T. (1944). "Transformations of Wiener Integrals under Translations".
*Annals of Mathematics*.**45**(2): 386–396. doi:10.2307/1969276. JSTOR 1969276. - 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.)
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- 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
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- 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
- 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.

**^**Oladyshkin S. and Nowak W. 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.**^**Oladyshkin S. and Nowak W. Incomplete statistical information limits the utility of high-order polynomial chaos expansions. Reliability Engineering & System Safety, 169, pp.137-148, 2018. DOI: 10.1016/j.ress.2012.05.002**^**Daniel Alpay and Alon Kipnis, Wiener Chaos Approach to Optimal Prediction, Numerical Functional Analysis and Optimization, 36:10, 1286-1306, 2015. DOI: 10.1080/01630563.2015.1065273