Tensorized multivariate basis enumeration functions

Enumeration functions (refer to Multivariate indices enumeration functions) help to enumerate a multivariate basis built as the tensorization of univariate basis, using the indexation of each marginal basis.

Such a multivariate basis is used, for example, in the functional chaos expansion setting.

Let consider some \inputDim univariate basis, denoted by (\pi_{k}^{(i)})_{k \geq 0} for 0 \leq i \leq \inputDim-1, where each \pi_{k}^{(i)}: \Rset \rightarrow \Rset.

Let denote by \{\psi_{\vect{\alpha}},\vect{\alpha} \in \Nset^\inputDim\} a multivariate basis built as the tensorization of the univariate basis. The multivariate basis term \psi_{\vect{\alpha}} is defined by the product:

\psi_{\vect{\alpha}} (\vect{\xi}) = \pi_{\alpha_0}^{(0)}(\xi_0) \times \dots \times \pi_{\alpha_{\inputDim-1}}^{({\inputDim-1})}(\xi_{\inputDim-1})

for any \vect{\xi} \in \mathbb{R}^{\inputDim}.

When the univariate basis are polynomials such that \alpha_i is the degree of \pi_{\alpha_i}^{(i)}, then the multi-index represents the marginal degrees of the polynomial \psi_{\vect{\alpha}}. In that case, the length of the multi-index is the total degree of the polynomial.

Several enumeration functions can be used:

  • the linear enumeration function,

  • the hyperbolic enumeration function,

  • the anisotropic hyperbolic enumeration function,

  • the infinity norm enumeration function.

We detail the interest of each one in the functional chaos expansion setting.

Linear enumeration function

The linear enumeration function builds the multivariate polynomials by increasing total degrees and for a given value of the total degree, by “graded reverse-lexicographic ordering” (see [sullivan2015]).

Hyperbolic enumeration function

The hyperbolic enumeration function is inspired by the so-called sparsity-of-effects principle, which states that most models are principally governed by main effects and low-order interactions. Accordingly, one wishes to define an enumeration function which first selects those multi-indices related to main effects, i.e. with a reasonably small number of nonzero components, prior to selecting those associated with higher-order interactions.

The hyperbolic enumeration functions are based on the q-norm defined in Multivariate indices enumeration functions, equation (1). The smaller q is, the more the indices containing high-order interactions are penalized.

The indices are ordered by increasing q-norm and for a same q-norm, according to the graded reverse-lexicographic ordering.

Anisotropic hyperbolic enumeration function

The anisotropic hyperbolic enumeration functions is a weighted version of the hyperbolic enumeration function, using a weighted q-norm defined in Multivariate indices enumeration functions, equation (3). The components with large anisotropic coefficients will have marginal degrees lower than the components with small anisotropic coefficients.

Therefore, when the model is governed by main effects of specific inputs, the associated weights should be small compared to the other ones.

The indices are ordered by increasing weighted q-norm and for a same weighted q-norm, according to the graded reverse-lexicographic ordering.

Infinity norm enumeration function

The infinity norm enumeration function is based on the infinity-norm defined in Multivariate indices enumeration functions, equation (4).

This function allows one to define the largest space of polynomials with given maximum marginal degree. When a tensorized Gaussian quadrature formula (GaussProductExperiment) is obtained by tensorization of univariate formulas, this space (with the appropriate marginal degrees) is the space of polynomials whose integrals are computed exactly by the quadrature rule.