Environmental Engineering Reference
In-Depth Information
6.3 PolYnoMIal ChaoS eXPanSIonS
6.3.1 Mathematical setting
Consider a computational model M whose input parameters are represented by a random
vector X , and the associated (random) QoI Y = M( X ). Assuming that Y has a finite variance
(which is a physically meaningful assumption when dealing with geotechnical systems), it
belongs to the so-called Hilbert space of second-order random variables, which allows for
the following representation (Soize and Ghanem, 2004):
0
Y
=
y Z
j
.
(6.3)
j
j
=
Z jj 0 is a
numerable set of random variables (which form a basis of the Hilbert space), and {}
In Equation 6.3 the random variable Y is cast as an infinite series, in which {}
y jj 0 are
coefficients. The latter may be interpreted as the coordinates of Y in this basis. Hilbertian
analysis guarantees the existence of such bases and representation; however, many choices
are possible. In the sequel, we focus on PCEs , in which the basis terms {}
Z jj 0 are multivari-
ate orthonormal polynomials in the input vector X , that is, Z j = Ψ j ( X ).
6.3.2 Construction of the basis
6.3.2.1 Univariate orthonormal polynomials
For the sake of simplicity, we assume that the input random vector has independent compo-
nents denoted by { X i , i = 1, …, M }, meaning that the joint distribution is simply the product
of the M marginal distributions {}:
f X M
i
=1
M
f
X ()
=
f
xx
Xi
( ,
D
,
(6.4)
i
X
i
i
i
=
1
where D X i is the support of X i . For each single variable X i and any two functions
φ 12
∈ D we deine a functional inner product by the following integral (pro-
vided it exists):
,
:
x
,
X i
= D
φφ φφ
12
,
(
xxfxdx
)()().
(6.5)
1
2
X
i
i
X i
Equation 6.5 is nothing but the expectation E[ϕ 1 ( X i ) ϕ 2 ( X i )] with respect to the marginal
distribution f X i . Two such functions are said to be orthogonal with respect to the prob-
ability measure P()
X = () if E[ϕ 1 ( X i ) ϕ 2 ( X i )] = 0. Using the above notation, classical
algebra allows one to build a family of orthogonal polynomials {
dx
fxdx
π k i
() , ∈ satisfying
k
}
def E
= D
()
i
()
i
()
i
()
i
()
i
()
i
ππ π
,
(
XX
)()
π
ππ
( )(
x
xxf
)()
xdxa
i
,
(6.6)
j
j
i
i
j
X
j
jk
k
k
k
i
X i
 
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