32
J. F. Reis et al.
Table 1.3 First order Sobol indices for each random variable of the germ w.r.t. the value of the
solution of Eq. (1.5) at the middle of the beam, u ∗ = u(x = 1/2/, ξ ). The space discretisation with
11 elements. Numerical values for a value of the mean μ = 1 and variance σ 2 = 0.1. The maximum
order of the orthogonal basis is N 0 = 2, which gives an expansion with 18,591 terms
l
0.016
0.08
0.4
2
10
S 1
1
3.835e − 01
4.761e − 01 9.535e − 01 9.996e − 01 9.999903e − 01
S 1
2
7.572e − 03
1.033e − 02 6.187e − 03 2.224e − 04 8.755946e − 06
S 1
3
4.610e − 01
4.483e − 01 3.829e − 02 5.235e − 05 8.145184e − 08
S 1
4
4.838e − 03
5.332e − 03 7.525e − 06 5.605e − 13 5.386858e − 20
S 1
5
6.338e − 02
3.673e − 02 1.332e − 06 4.395e − 11 1.173323e − 16
S 1
6
3.1801e − 03 1.403e − 03 2.411e − 10 5.364e − 23 1.067134e − 19
S 1
7
4.503e − 02
1.257e − 02 8.945e − 09 4.863e − 17 6.450506e − 20
S 1
8
3.712e − 03
3.772e − 04 6.991e − 16 3.020e − 21 3.854629e − 21
S 1
9
8.364e − 03
5.787e − 04 3.555e − 12 2.801e − 21 2.506504e − 26
S 1
10
3.540e − 03
5.780e − 05 9.602e − 23 2.181e − 26 3.183518e − 26
S 1
11
3.641e − 03
1.241e − 04 4.027e − 16 2.445e − 26 2.839943e − 26
1.7 Concluding Remarks
Physical phenomena are intrinsically affected by uncertainties. Therefore, the
mathematical models should also account for these uncertainties. In the chapter,
we provide an elementary example of such physical phenomena—heat diffusion
through a beam—and the corresponding model, Eq. (1.5). We illustrate a few UQ
questions with this example. More than the solution of the model, we are interested
in its QoI. We show two different ways of doing this: à la MC method and using SM.
Spectral methods are appropriate methods to perform these computations, given
their relatively cheap cost. In fact, complex models, such as the ones arising from
fluid dynamics, have many sources of uncertainties, and MC methods are generally
too demanding.
Heat diffusion through a beam may have different sources of uncertainty. In the
chapter, we only consider a random conductivity field, but other uncertainties may
be considered. We chose to parametrise the conductivity field, but other sampling
approaches could have been pursuit. The main reason why we chose to follow this
path is because, given the assumptions on the random field, the KL expansion
provides a parametrisation in independent random variables. This is important,
because this implies that the surrogate given by the PCE is also depending on
independent variables.
J. F. Reis et al.
Table 1.3 First order Sobol indices for each random variable of the germ w.r.t. the value of the
solution of Eq. (1.5) at the middle of the beam, u ∗ = u(x = 1/2/, ξ ). The space discretisation with
11 elements. Numerical values for a value of the mean μ = 1 and variance σ 2 = 0.1. The maximum
order of the orthogonal basis is N 0 = 2, which gives an expansion with 18,591 terms
l
0.016
0.08
0.4
2
10
S 1
1
3.835e − 01
4.761e − 01 9.535e − 01 9.996e − 01 9.999903e − 01
S 1
2
7.572e − 03
1.033e − 02 6.187e − 03 2.224e − 04 8.755946e − 06
S 1
3
4.610e − 01
4.483e − 01 3.829e − 02 5.235e − 05 8.145184e − 08
S 1
4
4.838e − 03
5.332e − 03 7.525e − 06 5.605e − 13 5.386858e − 20
S 1
5
6.338e − 02
3.673e − 02 1.332e − 06 4.395e − 11 1.173323e − 16
S 1
6
3.1801e − 03 1.403e − 03 2.411e − 10 5.364e − 23 1.067134e − 19
S 1
7
4.503e − 02
1.257e − 02 8.945e − 09 4.863e − 17 6.450506e − 20
S 1
8
3.712e − 03
3.772e − 04 6.991e − 16 3.020e − 21 3.854629e − 21
S 1
9
8.364e − 03
5.787e − 04 3.555e − 12 2.801e − 21 2.506504e − 26
S 1
10
3.540e − 03
5.780e − 05 9.602e − 23 2.181e − 26 3.183518e − 26
S 1
11
3.641e − 03
1.241e − 04 4.027e − 16 2.445e − 26 2.839943e − 26
1.7 Concluding Remarks
Physical phenomena are intrinsically affected by uncertainties. Therefore, the
mathematical models should also account for these uncertainties. In the chapter,
we provide an elementary example of such physical phenomena—heat diffusion
through a beam—and the corresponding model, Eq. (1.5). We illustrate a few UQ
questions with this example. More than the solution of the model, we are interested
in its QoI. We show two different ways of doing this: à la MC method and using SM.
Spectral methods are appropriate methods to perform these computations, given
their relatively cheap cost. In fact, complex models, such as the ones arising from
fluid dynamics, have many sources of uncertainties, and MC methods are generally
too demanding.
Heat diffusion through a beam may have different sources of uncertainty. In the
chapter, we only consider a random conductivity field, but other uncertainties may
be considered. We chose to parametrise the conductivity field, but other sampling
approaches could have been pursuit. The main reason why we chose to follow this
path is because, given the assumptions on the random field, the KL expansion
provides a parametrisation in independent random variables. This is important,
because this implies that the surrogate given by the PCE is also depending on
independent variables.