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
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
=δ
i
,
(6.6)
j
j
i
i
j
X
j
jk
k
k
k
i
X
i
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