1 Introduction to Spectral Methods for Uncertainty Quantification
29
σ
2
= σ
2
1 + σ
2
2 + σ
2
1,2 ,
(1.30)
where σ 2
1 and σ 2
2 are the contribution to the variance of ξ 1 and ξ 2 , respectively,
which are independent random variables. The term σ 2
1,2 represents a second order
contribution that accounts for combined interaction of the variables of the germ. In
this subsection, we are interested in the first order Sobol indices only, i.e. the values
of σ 2
1 and σ 2
2 .
1.6.4.1 MC Approach
If we ought to compute the values σ 2
1 and σ 2
2 using a MC approach, we would
need to sample M values of ξ 1 (being ξ 2 fixed) and compute the corresponding
M realisations. Then we could compute the variance of u ∗ for each ξ 1 , using the
corresponding samples. This gives σ 2
1 using the corresponding estimators [15]. A
similar procedure should be done for the second stochastic variable. An example
with M = 5 samples illustrated below.
1. Generate M samples of 4-dimensional points (2d, d = 2) in the unit hypercube
and construct the matrix M:
M =
⎛
⎝
0.3 0.3 0.7 0.3
0.2 0.3 0.8 0.4
0.2 0.1 0.5 0.4
⎞
⎠
2. Define the matrices A and B in the following way. The columns of A are the first
d columns of M. The columns of B are the remaining columns of M.
A =
⎛
⎝
0.3 0.3
0.2 0.3
0.2 0.1
⎞
⎠ B =
⎛
⎝
0.7 0.3
0.8 0.4
0.5 0.4
⎞
⎠
3. Construct matrices A
(i)
B , i = 1, . . . , M in the following way. The columns of
matrix A
(i)
B are the columns of matrix A except column i, which is the i-th
column of B.
A
(1)
B =
⎛
⎝
0.7 0.3
0.8 0.3
0.5 0.1
⎞
⎠ A
(2)
B =
⎛
⎝
0.7 0.3
0.8 0.3
0.5 0.1
⎞
⎠
4. Compute the M-dimensional vector U (A) , U (B) , U (i) , where each entry is the
solution of Eq. (1.15) using the points (ξ 1 , ξ 2 ) of the corresponding matrix. For
29
σ
2
= σ
2
1 + σ
2
2 + σ
2
1,2 ,
(1.30)
where σ 2
1 and σ 2
2 are the contribution to the variance of ξ 1 and ξ 2 , respectively,
which are independent random variables. The term σ 2
1,2 represents a second order
contribution that accounts for combined interaction of the variables of the germ. In
this subsection, we are interested in the first order Sobol indices only, i.e. the values
of σ 2
1 and σ 2
2 .
1.6.4.1 MC Approach
If we ought to compute the values σ 2
1 and σ 2
2 using a MC approach, we would
need to sample M values of ξ 1 (being ξ 2 fixed) and compute the corresponding
M realisations. Then we could compute the variance of u ∗ for each ξ 1 , using the
corresponding samples. This gives σ 2
1 using the corresponding estimators [15]. A
similar procedure should be done for the second stochastic variable. An example
with M = 5 samples illustrated below.
1. Generate M samples of 4-dimensional points (2d, d = 2) in the unit hypercube
and construct the matrix M:
M =
⎛
⎝
0.3 0.3 0.7 0.3
0.2 0.3 0.8 0.4
0.2 0.1 0.5 0.4
⎞
⎠
2. Define the matrices A and B in the following way. The columns of A are the first
d columns of M. The columns of B are the remaining columns of M.
A =
⎛
⎝
0.3 0.3
0.2 0.3
0.2 0.1
⎞
⎠ B =
⎛
⎝
0.7 0.3
0.8 0.4
0.5 0.4
⎞
⎠
3. Construct matrices A
(i)
B , i = 1, . . . , M in the following way. The columns of
matrix A
(i)
B are the columns of matrix A except column i, which is the i-th
column of B.
A
(1)
B =
⎛
⎝
0.7 0.3
0.8 0.3
0.5 0.1
⎞
⎠ A
(2)
B =
⎛
⎝
0.7 0.3
0.8 0.3
0.5 0.1
⎞
⎠
4. Compute the M-dimensional vector U (A) , U (B) , U (i) , where each entry is the
solution of Eq. (1.15) using the points (ξ 1 , ξ 2 ) of the corresponding matrix. For