Polynomial chaos
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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]