Polynomial chaos

Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms of a polynomial function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability distribution of these random variables. Note that despite its name, PCE has no immediate connections to chaos theory. The word "chaos" here should be understood as "random".[1]

PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables.[2] It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991[3] and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002.[4] Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G. Ernst and coworkers in 2011.[5]

PCE has found widespread use in engineering and the applied sciences because it makes possible to deal with probabilistic uncertainty in the parameters of a system. In particular, PCE has been used as a surrogate model to facilitate uncertainty quantification analyses.[6][7] PCE has also been widely used in stochastic finite element analysis[3] and to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters.[8]

  1. ^ The use of the word "chaos" by Norbert Wiener in his 1938 publication precedes the use of "chaos" in the branch of mathematics called chaos theory by almost 40 years. [1]
  2. ^ Wiener, Norbert (1938). "The Homogeneous Chaos". American Journal of Mathematics. 60 (4): 897–936. doi:10.2307/2371268. JSTOR 2371268.
  3. ^ a b Ghanem, Roger G.; Spanos, Pol D. (1991), "Stochastic Finite Element Method: Response Statistics", Stochastic Finite Elements: A Spectral Approach, New York, NY: Springer New York, pp. 101–119, doi:10.1007/978-1-4612-3094-6_4, ISBN 978-1-4612-7795-8, retrieved 2021-09-29
  4. ^ Xiu, Dongbin; Karniadakis, George Em (2002). "The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations". SIAM Journal on Scientific Computing. 24 (2): 619–644. Bibcode:2002SJSC...24..619X. doi:10.1137/s1064827501387826. ISSN 1064-8275. S2CID 10358251.
  5. ^ Ernst, Oliver G.; Mugler, Antje; Starkloff, Hans-Jörg; Ullmann, Elisabeth (2011-10-12). "On the convergence of generalized polynomial chaos expansions". ESAIM: Mathematical Modelling and Numerical Analysis. 46 (2): 317–339. doi:10.1051/m2an/2011045. ISSN 0764-583X.
  6. ^ Soize, Christian; Ghanem, Roger (2004). "Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure". SIAM Journal on Scientific Computing. 26 (2): 395–410. Bibcode:2004SJSC...26..395S. doi:10.1137/s1064827503424505. ISSN 1064-8275. S2CID 39569403.
  7. ^ O’Hagan, Anthony. "Polynomial chaos: A tutorial and critique from a statistician’s perspective." SIAM/ASA J. Uncertainty Quantification 20 (2013): 1-20.
  8. ^ "Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives]". IEEE Control Systems. 33 (5): 58–67. 2013. doi:10.1109/MCS.2013.2270410. ISSN 1066-033X. S2CID 5610154.