Environmental Engineering Reference
In-Depth Information
6.3.4.2 Projection
Owing to the orthogonality of the PC basis
(
Equation 6.11
),
one can compute each expan-
sion coefficient as follows:
δ
αβ
∑
∑
[
]
=
=
(6.24)
E
Y
Ψ
()
X
E
Ψ
()
X
⋅
y
Ψ
()
X
=
y
E
Ψ
() ()
X
Ψ
X
y
α
.
α
α
β
β
β
α
β
M
M
β∈
β∈
Thus, each coefficient
y
α
is nothing but the
orthogonal projection
of the random response
Y
onto the corresponding basis function Ψ
α
(
X
). The latter may be further elaborated as
[
]
=
∫
y
=
E
Ψ
Y
()
X
M
() () () .
x xxx
X
Ψ
f
d
(6.25)
α
α
α
D
X
The numerical estimation of
y
α
may be carried out with either one of the two expressions,
namely:
• By MCS allowing one to estimate the expectation in
Equation 6.25
(Ghiocel and
Ghanem, 2002). This technique however shows low efficiency, as does MCS in general;
• By the numerical integration of the right-hand side of
Equation 6.25
using Gaussian
quadrature (Le Maître et al., 2002; Berveiller et al., 2004a; Matthies and Keese, 2005).
The quadrature approach has been extended using
sparse grids
for a more efficient inte-
gration, especially in large dimensions. The so-called
stochastic collocation methods
have
also been developed. The reader is referred to the review paper by Xiu (2009) for more
details.
6.3.4.3 Least-square minimization
Instead of devising numerical methods that directly estimate each coefficient from the
expression
y
α
= E[
Y
Ψ
α
(
X
)], an alternative approach based on least-square minimization
and originally termed “regression approach” has been proposed in Berveiller et al. (2004b,
2006). The problem is set up as follows. Once a truncation scheme
A
⊂
M
is chosen (for
cated series and a residual:
∑
Y
=
M
()
X
=
y
X
αα
Ψε,
()
+
(6.26)
α∈
A
in which ε corresponds to all those PC polynomials whose index α is
not
in the truncation
set
A
. The least-square minimization approach consists of finding the set of coefficients
y
= {
y
α
, α ∈
A
} that minimizes the mean-square error
2
∑
def
E
[]
ε
2
=
E
Y
−
y
α
Ψ
X
()
,
(6.27)
α∈
A
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