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". 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 2011. 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. PCE has also been widely used in stochastic finite element analysis and to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters.
Main principles Polynomial chaos expansion (PCE) provides a way to represent a random variable Y {\displaystyle Y} with finite variance (i.e., Var ( Y ) < ∞ {\displaystyle \operatorname {Var} (Y)<\infty } ) as a function of an M {\displaystyle M} -dimensional random vector X {\displaystyle \mathbf {X} } , using a polynomial basis that is orthogonal with respect to the distribution of this random vector. The prototypical PCE can be written as:
Y = ∑ i ∈ N c i Ψ i ( X ) . {\displaystyle Y=\sum _{i\in \mathbb {N} }c_{i}\Psi _{i}(\mathbf {X} ).}
In this expression, c i {\displaystyle c_{i}} is a coefficient and Ψ i {\displaystyle \Psi _{i}} denotes a polynomial basis function. Depending on the distribution of X {\displaystyle \mathbf {X} } , different PCE types are distinguished.
Hermite polynomial chaos The original PCE formulation used by Norbert Wiener was limited to the case where X {\displaystyle \mathbf {X} } is a random vector with a Gaussian distribution. Considering only the one-dimensional case (i.e., M = 1 {\displaystyle M=1} and X = X {\displaystyle \mathbf {X} =X} ), the polynomial basis function orthogonal w.r.t. the Gaussian distribution are the set of i {\displaystyle i} -th degree Hermite polynomials H i {\displaystyle H_{i}} . The PCE of Y {\displaystyle Y} can then be written as:
Y = ∑ i ∈ N c i H i ( X ) {\displaystyle Y=\sum _{i\in \mathbb {N} }c_{i}H_{i}(X)} .
Generalized polynomial chaos Xiu generalized the result of Cameron–Martin to various continuous and discrete distributions using orthogonal polynomials from the so-called Askey-scheme and demonstrated L 2 {\displaystyle L_{2}} convergence in the corresponding Hilbert functional space. This is popularly known as the generalized polynomial chaos (gPC) framework. The gPC framework has been applied to applications including stochastic fluid dynamics, stochastic finite elements, solid mechanics, nonlinear estimation, the evaluation of finite word-length effects in non-linear fixed-point digital systems and probabilistic robust control. It has been demonstrated that gPC based methods are computationally superior to Monte-Carlo based methods in a number of applications. However, the method has a notable limitation. For large numbers of random variables, polynomial chaos becomes very computationally expensive and Monte-Carlo methods are typically more feasible.
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