Environmental Engineering Reference
In-Depth Information
XU
= T ( .
(6.13)
Depending on the marginal distribution of each input variable
X
i
, the associated reduced
variable
U
i
may be standard normal N (0, 1), standard uniform U(−1, 1), and so on. Then
the random model response
Y
is cast as a function of the reduced variables by composing the
computational model M and the transform T :
=
∑
Y
=
MMT
()
X
=
()
U
y
α
Ψ
α
( .
U
(6.14)
M
α∈
Note that the isoprobabilistic transform also allows one to address the case of correlated
input variables through, for example, Nataf transform (Ditlevsen and Madsen, 1996).
eXaMPLe 6.1
Suppose
X
= {
X
1
, …,
X
M
}
T
is a vector of independent Gaussian variables
X
i
~ N (μ
i
, σ
i
)
with the respective mean value μ
i
and standard deviation σ
i
. Then a one-to-one mapping
X
= T (
U
) is obtained by
X
=+ =…
µσ,
U
i
1
,
,
M
.
(6.15)
i
i
i
i
where
U
= {
U
1
, …,
U
M
}
T
is a standard normal vector.
eXaMPLe 6.2
Suppose
X
= {
X
1
,
X
2
}
T
where
X
1
~ LN (λ, ς) is a lognormal variable and
X
2
~ U(
a
,
b
) is a
uniform variable. It is natural to transform
X
1
into a standard normal variable and
X
2
into a standard uniform variable. The isoprobabilistic transform
X
= T (
U
) then reads:
‡
‰
‰
Xe
=
λς
U
1
1
(6.16)
ba
U
−
ba
+
X
=
+
2
2
2
2
6.3.3.2 Truncation scheme
The representation of the random response in
Equation 6.12
is exact when the infinite series
is considered. However, in practice, only a finite number of terms may be computed. For this
purpose, a
truncation scheme
has to be adopted. Since the polynomial chaos basis is made
of polynomials, it is natural to consider as a truncated series all polynomials up to a certain
degree. Let us define the
total degree
of a multivariate polynomial Ψ
α
by
M
∑
def
α=
α
i
.
(6.17)
i
=
1
The standard truncation scheme consists of selecting all polynomials such that |α| is
smaller than a given
p
, that is,
A
Mp
,
=∈ ≤
{
α
M
:
α
p
}.
(6.18)
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