Chaos expansions

Polynomial chaos, generalized polynomial chaos, arbitrary chaos etc


Placeholder, for a topic which has a slightly confusing name. To explore: Connection/difference from other methods of keeping track of evolution of uncertainty in dynamical systems. C&C Gaussian process regression as used in Gratiet, Marelli, and Sudret (2016), functional data analysis etc. This is not the same thing as chaos in the sense of the deterministic chaos made famous by dynamical systems theory and fractal t-shirts.

Polynomial chaos expansion

Wikipedia says:

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 R. Ghanem and Red-Horse (2017) and Kim et al. (2013), with divergent emphases on the history.

“Generalized”

Wikipedia credits Xiu (2010) with the particular generalisation which got naming rights, 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.

References

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.
Lei, Huan, Jing Li, Peiyuan Gao, Panos Stinis, and Nathan Baker. 2018. “A Data-Driven Framework for Sparsity-Enhanced Surrogates with Arbitrary Mutually Dependent Randomness,” April. https://doi.org/10.1016/j.cma.2019.03.014.
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.
Oladyshkin, S., and W. Nowak. 2012. “Data-Driven Uncertainty Quantification Using the Arbitrary Polynomial Chaos Expansion.” Reliability Engineering & System Safety 106 (October): 179–90. https://doi.org/10.1016/j.ress.2012.05.002.
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.
Wan, Xiaoliang, and George Em Karniadakis. 2006. “Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures.” SIAM Journal on Scientific Computing 28 (3): 901–28. https://doi.org/10.1137/050627630.
Wiener, Norbert. 1938. “The Homogeneous Chaos.” American Journal of Mathematics 60 (4): 897. https://doi.org/10.2307/2371268.
Witteveen, Jeroen A. S., and Hester Bijl. 2006. “Modeling Arbitrary Uncertainties Using Gram-Schmidt Polynomial Chaos.” In 44th AIAA Aerospace Sciences Meeting and Exhibit. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2006-896.
Xiu, Dongbin. 2010. Numerical Methods for Stochastic Computations: A Spectral Method Approach. USA: Princeton University Press. https://doi.org/10.2307/j.ctv7h0skv.
Zhang, Dongkun, Lu Lu, Ling Guo, and George Em Karniadakis. 2019. “Quantifying Total Uncertainty in Physics-Informed Neural Networks for Solving Forward and Inverse Stochastic Problems.” Journal of Computational Physics 397 (November): 108850. https://doi.org/10.1016/j.jcp.2019.07.048.
Zheng, Mengdi, Xiaoliang Wan, and George Em Karniadakis. 2015. “Adaptive Multi-Element Polynomial Chaos with Discrete Measure: Algorithms and Application to SPDEs.” Applied Numerical Mathematics 90 (April): 91–110. https://doi.org/10.1016/j.apnum.2014.11.006.