Polynomial chaos

rdf:langString Polynomial chaos

rdf:langString 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. PCE can be used, e.g., to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. Note that despite its name, PCE has no immediate connections to chaos theory. PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables. It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991 and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002. Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G. Ernst and coworkers in 2012. PCE has found widespread use in engineering and the applied sciences because it makes it possible to efficiently deal with probabilistic uncertainty in the parameters of a system. It is widely used in stochastic finite element analysis and as a surrogate model to facilitate uncertainty quantification analyses.