1 Introduction to Spectral Methods for Uncertainty Quantification
31
where Q q =
q
i=1
N
i
. As an example, consider N = 3 then m 3 = (0, 0, 1) ·
(ξ 1 , ξ 2 , ξ 3 ) = (0, 0, ξ 3 ), which means σ 2
m 3 := σ 2
3 which is the contribution of the
variable ξ 3 to the overall variance σ 2 . Finally, we just need to define the set of indices
s ∈ {1, . . . , P nisp } that are related to the variable(s) in m j , j = 1, . . . , Q N . Before
formally defining this multi-index, we look at the case of Table 1.1. Since N = 2, we
have 3 multi-indices m j , j = 1, 2, 3. One corresponds to ξ 1 , the other to ξ 2 and the
last to both of them. By construction, the polynomials Ψ s (ξ ) with index s = 1, 3, 6
depend only on ξ 1 ; the ones with index s = 2, 5, 10 depend only on ξ 2 ; and, finally,
the polynomials with indices s = 4, 7, 8 depend both on ξ 1 and ξ 2 . Therefore, we
can define the following three sets, according to the different dependences in the
germ:
S m 1 := {1, 3, 6}
S m 2 := {2, 5, 9}
S m 3 := {4, 7, 8}
The general definition of S m j is given by
S m j :=
s ∈ {1, .., P nisp } : Ψ s =
N
i=1
ψ m i (ξ i )
Now, we can write the decomposition of the variance, using this notation:
σ
2
=
Q N
j =1
σ
2
m j
(1.32)
where each σ 2
m j
is given w.r.t. the coefficients of the PCE in Eq. (1.18) as
σ
2
m j
:=
s∈S m j
u
2
s .
(1.33)
From Eq. (1.33) one can use the estimators as in [9, 16] to compute the Sobol indices
and even other quantitative indices. In Table 1.3, we can see the first order Sobol
indices of each variable ξ i for different correlation lengths l. We have the same
number of stochastic dimension as spatial elements, and therefore, we can clearly
appreciate their importance on the solution. For smaller correlation lengths, all the
stochastic variables have a significant importance on the QoI. This means that the
KL expansion should include all possible terms ξ i ; otherwise, the surrogate will
not be accurate. On the other hand, if the correlation length is large, then the only
significant variable is the ξ i . This is because the only significant mode in the KL
expansion is exactly the first one.
31
where Q q =
q
i=1
N
i
. As an example, consider N = 3 then m 3 = (0, 0, 1) ·
(ξ 1 , ξ 2 , ξ 3 ) = (0, 0, ξ 3 ), which means σ 2
m 3 := σ 2
3 which is the contribution of the
variable ξ 3 to the overall variance σ 2 . Finally, we just need to define the set of indices
s ∈ {1, . . . , P nisp } that are related to the variable(s) in m j , j = 1, . . . , Q N . Before
formally defining this multi-index, we look at the case of Table 1.1. Since N = 2, we
have 3 multi-indices m j , j = 1, 2, 3. One corresponds to ξ 1 , the other to ξ 2 and the
last to both of them. By construction, the polynomials Ψ s (ξ ) with index s = 1, 3, 6
depend only on ξ 1 ; the ones with index s = 2, 5, 10 depend only on ξ 2 ; and, finally,
the polynomials with indices s = 4, 7, 8 depend both on ξ 1 and ξ 2 . Therefore, we
can define the following three sets, according to the different dependences in the
germ:
S m 1 := {1, 3, 6}
S m 2 := {2, 5, 9}
S m 3 := {4, 7, 8}
The general definition of S m j is given by
S m j :=
s ∈ {1, .., P nisp } : Ψ s =
N
i=1
ψ m i (ξ i )
Now, we can write the decomposition of the variance, using this notation:
σ
2
=
Q N
j =1
σ
2
m j
(1.32)
where each σ 2
m j
is given w.r.t. the coefficients of the PCE in Eq. (1.18) as
σ
2
m j
:=
s∈S m j
u
2
s .
(1.33)
From Eq. (1.33) one can use the estimators as in [9, 16] to compute the Sobol indices
and even other quantitative indices. In Table 1.3, we can see the first order Sobol
indices of each variable ξ i for different correlation lengths l. We have the same
number of stochastic dimension as spatial elements, and therefore, we can clearly
appreciate their importance on the solution. For smaller correlation lengths, all the
stochastic variables have a significant importance on the QoI. This means that the
KL expansion should include all possible terms ξ i ; otherwise, the surrogate will
not be accurate. On the other hand, if the correlation length is large, then the only
significant variable is the ξ i . This is because the only significant mode in the KL
expansion is exactly the first one.