Polynomial Chaos Expansion - PCE

Polynomial Chaos Expansions (PCE) represent a class of methods which employ orthonormal polynomials to construct approximate response surfaces (metamodels or surrogate models) to identify a mapping between inputs and outputs of a numerical model [58]. PolynomialChaosExpansion methods can be directly used for moment estimation and sensitivity analysis (Sobol indices). A PCE object can be instantiated from the class PolynomialChaosExpansion. The method can be used for models of both one-dimensional and multi-dimensional outputs.

Let us consider a computational model \(Y = \mathcal{M}(x)\), with \(Y \in \mathbb{R}\) and a random vector with independent components \(X \in \mathbb{R}^M\) described by the joint probability density function \(f_X\). The polynomial chaos expansion of \(\mathcal{M}(x)\) is

\[Y = \mathcal{M}(x) = \sum_{\alpha \in \mathbb{N}^M} y_{\alpha} \Psi_{\alpha} (X)\]

where the \(\Psi_{\alpha}(X)\) are multivariate polynomials orthonormal with respect to \(f_X\) and \(y_{\alpha} \in \mathbb{R}\) are the corresponding coefficients.

Practically, the above sum needs to be truncated to a finite sum so that \(\alpha \in A\) where \(A \subset \mathbb{N}^M\). The polynomial basis \(\Psi_{\alpha}(X)\) is built from a set of univariate orthonormal polynomials \(\phi_j^{i}(x_i)\) which satisfy the following relation

\[\Big< \phi_j^{i}(x_i),\phi_k^{i}(x_i) \Big> = \int_{D_{X_i}} \phi_j^{i}(x_i),\phi_k^{i}(x_i) f_{X_i}(x_i)dx_i = \delta_{jk}\]

The multivariate polynomials \(\Psi_{\alpha}(X)\) are assembled as the tensor product of their univariate counterparts as follows

\[\Psi_{\alpha}(X) = \prod_{i=1}^M \phi_{\alpha_i}^{i}(x_i)\]

which are also orthonormal.