Polynomial chaos basis¶
has a finite variance, i.e. ;
has independent components.
where the ’s are multivariate polynomials that are orthonormal with respect to the joint PDF , that is:
where if and 0 otherwise, and the ’s are deterministic coefficients that fully characterize the response .
Common choices for are standard distributions such as a standard normal distribution or a uniform distribution over . For simplicity, it is assumed from now on that the components of the original input random vector have been already scaled, i.e. .
where is the marginal PDF of . Let us consider a family of orthonormal polynomials with respect to , :
The reader is referred to for details on the selection of suitable families of orthogonal polynomials. It is assumed that the degree of is for and (). Upon tensorizing the resulting families of univariate polynomials, one gets a set of orthonormal multivariate polynomials defined by:
where the multi-index notation has been introduced.
Building of the PC basis – dependent random variables
where is a function of the copula of . Note that such a basis is no longer polynomial. When dealing with independent random variables, one gets and each basis element may be recast as in (5). Determining is usually computationally expensive though, hence an alternative strategy for specific types of input random vectors.
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 by a polynomial approximation over its whole domain of definition as shown in . Actually this approach is strictly equivalent to:
Regarding the input parameters as random uniform random variables
Expanding any quantity of interest provided by the model onto a PC expansion made of Legendre polynomials
See the available orthogonal basis.
Ghanem and P. Spanos, 1991, “Stochastic finite elements – A spectral approach”, Springer Verlag. (Reedited by Dover Publications, 2003).