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.
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.
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