Polynomial chaos basis

The current section is focused on a specific kind of functional chaos representation that has been implemented, namely polynomial chaos expansions.
Mathematical framework
Throughout this section, the model response is assumed to be a scalar random variable Y = h(\underline{X}). However, the following derivations hold in case of a vector-valued response.
Let us suppose that:
  • Y = h(\underline{X}) has a finite variance, i.e. \Var{h(\underline{X})} < \infty;
  • \underline{X} has independent components.
Then it is shown that \underline{Y} may be expanded onto the PC basis as follows:

(1)Y \, \,  \equiv \, \,  h(\underline{X}) \, \, = \, \, \sum_{j=0}^{\infty} \; a_{j} \; \psi_{j}(\underline{X})

where the \psi_{j}’s are multivariate polynomials that are orthonormal with respect to the joint PDF f_{\underline{X}}(\underline{x}), that is:

\langle \psi_{j}(\underline{X}) \; , \; \psi_{k}(\underline{X}) \rangle \, \, \equiv \, \, \Expect{\psi_{j}(\underline{X}) \; \psi_{k}(\underline{X})} \, \, = \, \, \delta_{j,k}

where \delta_{j,k} = 1 if j=k and 0 otherwise, and the a_{j}’s are deterministic coefficients that fully characterize the response \underline{Y}.

Building of the PC basis – independent random variables
We first consider the case of independent input random variables. In practice, the components X_i of random vector \underline{X} are rescaled using a specific mapping T_i, usually referred to as an isoprobabilistic transformation (see ). The set of scaled random variables reads:

(2)U_i \, \, = \, \, T_i(X_i) \quad \quad , \quad \, i=1,\dots,n

Common choices for U_i are standard distributions such as a standard normal distribution or a uniform distribution over [-1,1]. For simplicity, it is assumed from now on that the components of the original input random vector \underline{X} have been already scaled, i.e. X_i = U_i.

Let us first notice that due to the independence of the input random variables, the input joint PDF may be cast as:

(3)f_{\vect{X}}(\vect{x}) \, = \, \prod_{i=1}^{n} f_{X_i}(x_{i})

where f_{X_i}(x_{i}) is the marginal PDF of X_i. Let us consider a family \{\pi^{(i)}_{j}, j \in \Nset\} of orthonormal polynomials with respect to f_{X_i}, :

(4)\langle \pi^{(i)}_{j}(X_{i}) \; , \; \pi^{(i)}_{k}(X_{i}) \rangle  \, \, \equiv \, \, \Expect{\pi^{(i)}_{j}(X_{i}) \;  \pi^{(i)}_{k}(X_{i})} \, \, = \, \, \delta_{j,k}

The reader is referred to  for details on the selection of suitable families of orthogonal polynomials. It is assumed that the degree of \pi^{(i)}_{j} is j for j>0 and \pi^{(i)}_{0} \equiv 1 (i=1,\dots,n). Upon tensorizing the n resulting families of univariate polynomials, one gets a set of orthonormal multivariate polynomials \{\psi_{\idx}, \idx \in \Nset^n\} defined by:

(5)\psi_{\idx}(\vect{x}) \, \, \equiv \,\, \pi^{(1)}_{\alpha_{1}}(x_{1}) \times \cdots \times \pi^{(n)}_{\alpha_{n}}(x_{n})

where the multi-index notation \idx \equiv \{\alpha_{1},\dots,\alpha_{M}\} has been introduced.

Building of the PC basis – dependent random variables

In case of dependent variables, it is possible to build up an orthonormal basis as follows:

(6)\psi_{\idx}(\vect{x}) \, \, = \,\,  K(\underline{x}) \;\prod_{i=1}^M \pi^{(i)}_{\alpha_{i}}(x_{i})

where K(\underline{x}) is a function of the copula of \vect{X}. Note that such a basis is no longer polynomial. When dealing with independent random variables, one gets K(\underline{x})=1 and each basis element may be recast as in (5). Determining K(\underline{x}) is usually computionally expensive though, hence an alternative strategy for specific types of input random vectors.

If \vect{X} has an elliptical copula instead of an independent one, it may be recast as a random vector \vect{U} with independent components using a suitable mapping T : \vect{X} \mapsto \vect{U} such as the Nataf transformation. The so-called Rosenblatt transformation may also be applied in case of a Gaussian copula . Thus the model response Y may be regarded as a function of \vect{U} and expanded onto a polynomial chaos expansion with basis elements cast as in (5).
Link with classical deterministic polynomial approximation

In a deterministic setting (i.e. when the input parameters are considered to be deterministic), it is of common practice to substitute the model function h by a polynomial approximation over its whole domain of definition as shown in . Actually this approach is strictly equivalent to:

  1. Regarding the input parameters as random uniform random variables
  2. Expanding any quantity of interest provided by the model onto a PC expansion made of Legendre polynomials

API:

References:

    1. Ghanem and P. Spanos, 1991, “Stochastic finite elements – A spectral approach”, Springer Verlag. (Reedited by Dover Publications, 2003).