Environmental Engineering Reference
In-Depth Information
Problems involving spatial variability of the material properties may be efficiently
addressed by introducing random fields and using discretization techniques. The discretized
field is represented by a vector of (usually) standard normal variables, whose size may be
large (e.g., 20-100), especially when the correlation length of the field is small. However, the
so-called sparse PC expansion technique may be used so that the full analysis can be carried
out using a few hundred to a thousand model runs.
aCknoWleDgMentS
The author thanks Dr. Géraud Blatman (EDF R&D, France) for the original joint work
on sparse PC expansions and ongoing discussions. The support of Dr. Stefano Marelli and
Roland Schöbi (ETH Zürich, Chair of Risk, Safety, and Uncertainty Quantification) for
preparing the numerical examples is also gratefully acknowledged.
aPPenDIX 6a: herMIte PolYnoMIalS
The Hermite polynomials
He
n
(
x
) are the solution of the following differential equation:
y
″ −
xy
′ +
ny
′ = 0.
n
∈ N
(A.1)
They may be generated in practice by the following recurrence relationship:
He
0
(
x
) = 1.
(A.2)
He
n
+1
(
x
) =
x He
n
(
x
) −
n He
n
−1
(
x
).
(A.3)
They are orthogonal with respect to the Gaussian probability measure:
∞
∫
He
()
xHex xdxn
()()
φ
=
!
δ
,
(A.4)
m
n
mn
−∞
2
2
π
/
is the standard normal PDF. If
U
is a standard normal random
variable, the following relationship holds:
−
x
where φ(
x
=
12
e
E
[
He
m
(
U
)
He
n
(
U
)] =
n
!δ
mn.
(A.5)
The first four Hermite polynomials are
He
0
(
x
) = 1
He
1
(
x
) =
x He
2
(
x
) =
x
2
− 1
He
3
(
x
) =
x
3
− 3
x.
(A.6)
Owing to Equation A.5, the orthonormal polynomials read:
2
3
ψ
()
x
=
1
ψ
()
x
= =− =−
x
ψ
()
x
(
x
1
)
2
ψ
()
x
(
xx
3
)
6
.
(A.7)
0
1
2
3
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