Environmental Engineering Reference
In-Depth Information
The number of terms in the truncated series is
Mp
p
+
=
(
Mp
Mp
+
)!
!!
.
card A
M,p
=
(6.19)
The maximal polynomial degree
p
may typically be equal to 3-5 in practical applications.
The question on how to define the suitable
p
for obtaining a given accuracy in the truncated
series will be addressed later in Section 6.3.6. Note that the cardinality of
A
M,p
increases
polynomially with
M
and
p
. Thus, the number of terms in the series, that is, the number of
coefficients to be computed, increases dramatically, say when
M
> 10. This complexity is
referred to as the
curse of dimensionality.
Other advanced truncation schemes that allow
one to bypass this problem will be considered later on in Section 6.3.6. As a conclusion, the
construction of a truncated PC expansion requires to
• Transform the input random vector
X
into reduced variables;
• Compute the associated families of univariate orthonormal polynomials;
• Compute the set of multi-indices corresponding to the truncation set
(
Equation 6.18
).
For this purpose, two different algorithms may be found in Sudret et al. (2006,
Appendix I) and Blatman (2009, Appendix C).
6.3.3.3 Application example
Let us consider a computational model
y
=
M
(
x
1
,
x
2
) involving two random parameters {
X
1
,
X
2
} that are modeled by
lognormal
distributions, for example, the load-carrying capac-
ity of a foundation in which the soil cohesion and friction angle are considered uncertain.
Denoting by (λ
i
, ζ
i
) the parameters of each distribution (i.e., the mean and standard devia-
tion of the logarithm of
X
i
, i
= 1, 2), the input variables may be transformed into reduced
standard normal variables
X
= T (
U
) as follows:
X
∼
∼
LN
LN
(,)
(,)
λζ
λζ
X
=
=
e
λζ
+
U
111
1
1
1
1
.
(6.20)
X
X
e
λζ
+
U
222
2
2
2
2
The problem reduces to representing a function of two standard normal variables onto a
PCE:
)
=
∑
(
Y
=
MT(,
U
U
)
y
Ψ
(
UU
,
.
(6.21)
12
α
12
2
α∈
Since the reduced variables are standard normal, Hermite polynomials are used. Their
derivation is presented in detail in Appendix A. For the sake of illustration, the orthonormal
Hermite polynomials up to degree
p
= 3 read (see
Equation 6.8
)
:
2
3
(6.22)
ψ
0
()
x
=
1
ψ
()
x
= =− =−
x
ψ
()
x
(
x
1
)
2
ψ
()
x
(
xx
3
)
6
.
1
2
3
Suppose a standard truncation scheme of maximal degree
p
= 3 is selected. This leads to
a truncation set
A
2,3
of size
P
=
23
10
The set of multi-indices (α
1
, α
2
) such that {α
i
≥ 0,
polynomials.
()
=
.
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