Polynomial chaos basisΒΆ

This page introduces polynomial chaos expansion. We consider the notations introduced in Functional Chaos Expansion.

Let T : \Rset^{n_X} \rightarrow \Rset^{n_X} be the isoprobabilistic transformation such that \vect{Z} = T(\vect{X}) has independent marginals. Therefore, the probability density function of the standard random vector \vect{Z} is the product of the marginal probability density functions:

\mu_{\vect{Z}}(\vect{z})= \prod_{i=1}^{n_X} \mu_{Z_i}(z_i)

for any \vect{z} \in \Rset^{n_X} where \mu_{\vect{Z}} is the joint PDF of the random vector \vect{Z} and \mu_{Z_i} is the marginal PDF of the random variable Z_i.

For any i \in \{0, ..., n_X\}, let \left(\pi_k^{(i)}\right)_{k \geq 0} be the family of univariate polynomials of the i-th marginal orthogonal with respect to the marginal distribution \mu_{Z_i}, where k represents the polynomial degree. The multivariate polynomial basis can be built using the tensor product of univariate polynomials which are orthonormal with respect to \mu_{\vect{Z}}. The orthonormal polynomials are:

\Psi_\vect{\alpha}(\vect{z}) = \prod_{i=1}^{n_X} \pi_{\alpha_i}^{(i)}(z_i)

where \vect{\alpha} = (\alpha_1, \dots, \alpha_{n_X}) \in \Nset^{n_X} is the multi-index representing the marginal polynomial degrees and \pi_{\alpha_i}^{(i)} is the i-th marginal univariate orthonormal polynomial of degree \alpha_i. The orthonormal polynomial basis with respect to the marginal \mu_{Z_i} is known for some distributions: see Orthogonal polynomials for more details on classical orthonormal polynomial families.

For classical polynomials, the StandardDistributionPolynomialFactory class implements the three-term recurrence. If the family is not already known, the polynomials can be represented by their three-term recurrence, using the adaptive Stieljes algorithm (see AdaptiveStieltjesAlgorithm). Once the sequence of recurrence coefficients is known, the reverse Clenshaw algorithm enables fast, stable evaluation of the polynomials at any point (see OrthogonalUniVariatePolynomial).