30
J. F. Reis et al.
instance,
U
(1)
=
⎡
⎢
⎣
u(x, ξ
(1)
1 = 0.7, ξ
(1)
2 = 0.3)
u(x, ξ
(2)
1 = 0.8, ξ
(2)
2 = 0.3)
u(x, ξ
(3)
1 = 0.5, ξ
(3)
2 = 0.1)
⎤
⎥
⎦
5. Use the estimators [15] to compute the Sobol indices. For instance, the first order
Sobol indices are given by
S i =
V i
V
(1.31)
where V is some approximation of σ 2 and
V i =
1
N
M
j =1
U
(B)
j
U
(i)
j − U
(A)
j
.
It is clear that this turns out to be a quite computationally demanding algorithm
when the number of samples M is high. Indeed, we need to find M(d + 2)
realisations to compute Eq. (1.31). We saw in Sect. 1.5 that the number of samples
needed is of the order of thousands. In the next subsection, we present a different
approach to compute the Sobol indices using the PCE as a surrogate.
1.6.4.2 Surrogate Approach
There are many ways of computing the Sobol indices using a surrogate model. In the
following, we shall consider a PCE obtained using the NISP method. We notice that
the coefficients are obtained using a quadrature rule. We also highlighted that the
number of realisations of the solution is related to the number of quadrature points
used. For a sufficient accurate Gauss–Hermite quadrature rule, the number of points
should be much less than the number of realisations M previously used in the MC
approach; therefore, the number of realisations for NISP should be much less than
M.
In this subsection, we give an explicit formula to compute the Sobol indices. We
also illustrate the meaning of these coefficients with a numerical example. Consider
the multi-index m j = α j · ξ , where the multi-index α j ∈ {0, 1} N is given by
α 1 = (1, 0, 0, · · · , 0) α Q 1 +1 = (1, 1, 0, · · · , 0) · · · α Q N = (1, · · · , 1, 1, 1)
α 2 = (0, 1, 0, · · · , 0) α Q 1 +2 = (1, 0, 1, · · · , 0) · · ·
. . .
. . .
α Q 1 = (0, 0, 0, · · · , 1) α Q 2 = (1, 0, 0, · · · , 1) · · ·
J. F. Reis et al.
instance,
U
(1)
=
⎡
⎢
⎣
u(x, ξ
(1)
1 = 0.7, ξ
(1)
2 = 0.3)
u(x, ξ
(2)
1 = 0.8, ξ
(2)
2 = 0.3)
u(x, ξ
(3)
1 = 0.5, ξ
(3)
2 = 0.1)
⎤
⎥
⎦
5. Use the estimators [15] to compute the Sobol indices. For instance, the first order
Sobol indices are given by
S i =
V i
V
(1.31)
where V is some approximation of σ 2 and
V i =
1
N
M
j =1
U
(B)
j
U
(i)
j − U
(A)
j
.
It is clear that this turns out to be a quite computationally demanding algorithm
when the number of samples M is high. Indeed, we need to find M(d + 2)
realisations to compute Eq. (1.31). We saw in Sect. 1.5 that the number of samples
needed is of the order of thousands. In the next subsection, we present a different
approach to compute the Sobol indices using the PCE as a surrogate.
1.6.4.2 Surrogate Approach
There are many ways of computing the Sobol indices using a surrogate model. In the
following, we shall consider a PCE obtained using the NISP method. We notice that
the coefficients are obtained using a quadrature rule. We also highlighted that the
number of realisations of the solution is related to the number of quadrature points
used. For a sufficient accurate Gauss–Hermite quadrature rule, the number of points
should be much less than the number of realisations M previously used in the MC
approach; therefore, the number of realisations for NISP should be much less than
M.
In this subsection, we give an explicit formula to compute the Sobol indices. We
also illustrate the meaning of these coefficients with a numerical example. Consider
the multi-index m j = α j · ξ , where the multi-index α j ∈ {0, 1} N is given by
α 1 = (1, 0, 0, · · · , 0) α Q 1 +1 = (1, 1, 0, · · · , 0) · · · α Q N = (1, · · · , 1, 1, 1)
α 2 = (0, 1, 0, · · · , 0) α Q 1 +2 = (1, 0, 1, · · · , 0) · · ·
. . .
. . .
α Q 1 = (0, 0, 0, · · · , 1) α Q 2 = (1, 0, 0, · · · , 1) · · ·