Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 266

3

He 1 ( x )
He 2 ( x )
He 3 ( x )
He 4 ( x )

2

1

0

-1

-2

-3 -3

-2 -1 0 1 2 3

x

Figure 6.2 Univariate Hermite polynomials.

ψ k i k ∈ N are defined according to the i ith marginal
distribution, see Equations 6.6 through 6.8 . By virtue of Equation 6.6 a nd the above tensor
product construction, the multivariate polynomials in the input vector X are also orthonor-
mal, that is,

()

where the univariate polynomials { , }

def

∫

= (6.11)

E Ψ α () () XX xxxx

X

Ψ Ψ () () () Ψ f d = δ ∀α, β ∈ N

M

,

β α β αβ

D X

where δ αβ is the Kronecker symbol that is equal to 1 if α = β and zero otherwise. With this
notation, it can be proven that the set of all multivariate polynomials in the input random
vector X forms a basis of the Hilbert space in which Y = M( X ) is to be represented (Soize
and Ghanem, 2004):

= ∑ α Ψ α
α∈ N

Y y ( . X

(6.12)

M

This equation may be interpreted as an intrinsic representation of the random response Y
in an abstract space through an orthonormal basis and coefficients that are the coordinates
of Y in this basis.

6.3.3 Practical Implementation
6.3.3.1 Isoprobabilistic transform
In practical uncertainty quantification problems, the random variables that model the input
parameters (e.g., material properties, loads, etc.) are usually not standardized as those
shown in Table 6.1 . Thus, it is necessary to first transform the random vector X into a set of
reduced variables U through an isoprobabilistic transform