polynomial_chaos.rst

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 PCE1. .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 = ℳ(x), with Y ∈ ℝ and a random vector with independent components X ∈ ℝ^M described by the joint probability density function f_X. The polynomial chaos expansion of ℳ(x) is

Y = ℳ(x) = ∑_{α ∈ ℕ^M}y_αΨ_α(X)

where the Ψ_α(X) are multivariate polynomials orthonormal with respect to f_X and y_α ∈ ℝ are the corresponding coefficients.

Practically, the above sum needs to be truncated to a finite sum so that α ∈ A where A ⊂ ℕ^M. The polynomial basis Ψ_α(X) is built from a set of univariate orthonormal polynomials ϕ_jⁱ(x_i) which satisfy the following relation

<ϕ_jⁱ(x_i), ϕ_kⁱ(x_i)> = ∫_DXiϕ_jⁱ(x_i), ϕ_kⁱ(x_i)f_Xi(x_i)dx_i = δ_jk

The multivariate polynomials Ψ_α(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.

Polynomial Bases <pce/polynomial_bases> Polynomials <pce/polynomials> Regressions <pce/regressions> Polynomial Chaos Expansion <pce/pce> Physics-informed Polynomial Chaos Expansion <pce/physics_informed>