1 Polynome d'Hermite

$$ H_{0}(\xi_{1}) = 1 $$

$$ H_{0}(\xi_{2}) = 1 $$

$$ H_{1}(\xi_{1}) = \xi_{1} $$

$$ H_{1}(\xi_{2}) = \xi_{1} $$

2 Bi dimentionelle

$$ \psi_{0}(\xi_{1}, \xi_{2}) = H_{0}(\xi_{1})H_{0}(\xi_{2}) = 1 $$

$$ \psi_{1}(\xi_{1}, \xi_{2}) = H_{1}(\xi_{1})H_{0}(\xi_{2}) = \xi_{1} $$

$$ \psi_{2}(\xi_{1}, \xi_{2}) = H_{0}(\xi_{1})H_{1}(\xi_{2}) = \xi_{2} $$

3 Ishigami

On calcule une troncature de dimension d et d'ordre p en suivant la formule suivante (avec p = 3)

$$ f_{p}(\xi_{1}, \xi_{2}, \xi_{3}) = \sum_{k = 0}^{P}\alpha_{k}\psi_{k}(\xi_{1}, \xi_{2}, \xi_{3}) $$

$$ P+1 = \frac{(p+d)!}{p!d!} $$

$$ \alpha_{k} = MC $$