# Polynomial chaos expansion

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## Vanilla

Polynomial chaos (PC), also called Wiener chaos expansion,is a non-sampling-based method to determine evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. PC was first introduced by Norbert 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 $$L_2$$ sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems.

There are friendly formal introductions in Ghanem and Red-Horse (2017) and Kim et al. (2013), with divergent emphases on the history.

To explore: Connection/difference from to kernel tricks and the use of covariance kernels and Gaussian process regression as used in Gratiet, Marelli, and Sudret (2016).

To answer: Is this precisely the Karhunen—Loève expansion trick? Is anything special going on here?

I’m reading a little further on this; It looks very similar to the set up of functional data analysis. I wonder if there is any distinction at all apart from terminology? TBD.

## Generalized

Wikipedia credits Xiu (2010) with this, based off Cameron-Martin formulae for Wiener measures about which I know nothing but looking at the context I feel like I might have missed a run there.

Alexanderian, Alen. 2015. “A Brief Note on the Karhunen-Loève Expansion.” October 26, 2015. http://arxiv.org/abs/1509.07526.

Calatayud Gregori, Julia, Benito M. Chen-Charpentier, Juan Carlos Cortés López, and Marc Jornet Sanz. 2019. “Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models.” Symmetry 11 (1): 43. https://doi.org/10.3390/sym11010043.

Franceschini, Chiara, and Cristian Giardinà. 2017. “Stochastic Duality and Orthogonal Polynomials.” January 31, 2017. http://arxiv.org/abs/1701.09115.

Ghanem, Roger G., and Pol D. Spanos. 2003. Stochastic Finite Elements: A Spectral Approach. New York, NY: Courier Corporation. https://doi.org/10.1007/978-1-4612-3094-6_1.

Ghanem, Roger, and John Red-Horse. 2017. “Polynomial Chaos: Modeling, Estimation, and Approximation.” In Handbook of Uncertainty Quantification, edited by Roger Ghanem, David Higdon, and Houman Owhadi, 521–51. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-12385-1_13.

Gratiet, Loïc Le, Stefano Marelli, and Bruno Sudret. 2016. “Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes.” In Handbook of Uncertainty Quantification, edited by Roger Ghanem, David Higdon, and Houman Owhadi, 1–37. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-11259-6_38-1.

Kim, Kwang-Ki K., and Richard D. Braatz. 2013. “Generalised Polynomial Chaos Expansion Approaches to Approximate Stochastic Model Predictive Control .” International Journal of Control 86 (8): 1324–37. https://doi.org/10.1080/00207179.2013.801082.

Kim, Kwang-Ki K., Dongying Erin Shen, Zoltan K. Nagy, and Richard D. Braatz. 2013. “Wiener’s Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives].” IEEE Control Systems Magazine 33 (5): 58–67. https://doi.org/10.1109/MCS.2013.2270410.

Levajkovic, Tijana, and Dora Selesi. 2011. “Chaos Expansion Methods for Stochastic Differential Equations Involving the Malliavin Derivative, Part I.” Publications de L’Institut Mathematique 90 (104): 65–84. https://doi.org/10.2298/PIM1104065L.

Luo, Wuan. 2006. “Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations.” PhD thesis, California Institute of Technology. https://doi.org/10.7907/RPKX-BN02.

Nualart, David, and Wim Schoutens. 2000. “Chaotic and Predictable Representations for Lévy Processes.” Stochastic Processes and Their Applications 90 (1): 109–22. https://doi.org/10.1016/S0304-4149(00)00035-1.

Schoutens, Wim. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics. New York: Springer-Verlag. https://doi.org/10.1007/978-1-4612-1170-9.

Wiener, Norbert. 1938. “The Homogeneous Chaos.” American Journal of Mathematics 60 (4): 897. https://doi.org/10.2307/2371268.

Xiu, Dongbin. 2010. Numerical Methods for Stochastic Computations: A Spectral Method Approach. USA: Princeton University Press. https://doi.org/10.2307/j.ctv7h0skv.