28
J. F. Reis et al.
Algorithm 3 Stochastic Galerkin algorithm to compute the PCE coefficients
1 Compute the coefficients k s (x) of the expansion in Eq. (1.26).
2 Compute matrices M s for each k s (x).
3 Compute the tensor C in Eq. (1.27).
4 Compute the block matrices in Eq. (1.28).
5 Solve the linear system of Eq. (1.29).
One way of computing the coefficients k s (x) is by using the KL expansion of
k(x, ξ ). This means we need to write the expansion in Eq. (1.13) in the form of
Eq. (1.26), and then we can use these coefficients to build the matrices M s . However,
there are ways of computing these matrices directly from the KL expansion of
k(x, ξ ); see [9]. Next, we need to compute the third order tensor C that has
(P G + 1) 3 entries. We can do so by exploiting the orthogonality properties of
the polynomials Ψ s (x). This yields a symmetric and sparse tensor that makes the
procedure much more efficient. Finally, after completing step 4 we solve the linear
system in Eq. (1.29). This is a sparse system of size N el (P G + 1) × N el (P G + 1).
Again, there are a number of techniques one could exploit to solve this system
in a more efficient way. These include the use of Krylov-based methods and preconditioning techniques [9].
1.6.4 Application of Surrogate Models: A Sensitivity Analysis
Using PC Expansions
We can exploit the coefficients of the PCE of a model to get statistical information
about the QoI, and this is a great improvement in terms of efficiency, if we were
to use a MC approach, instead. In this section, we discuss the computation of the
Sobol indices, a set of parameter which is very useful for sensitivity analysis. We
do not provide a detailed explanation about how to compute Sobol indices, the
interested reader can refer to [15] for details on this. The goal is to present a general
overview about differences in computing the Sobol indices using a MC approach
or a surrogate model approach. We also compute the Sobol indices for a specific
example, using the surrogate model approach, as in [9, 16].
Let ξ := {(ξ 1 , . . . , ξ N } be a sample of the germ, and consider the solution value
at the midpoint of the beam for this particular sample, i.e. u ∗ = u(x = 1/2, ξ). We
are interested in the significance of each random variable ξ i alone with respect to
the value u ∗ . This information is provided by the first order Sobol indices.
Let σ 2 be the variance of u ∗ . This variance accounts for the variability of all
random variables ξ i that contributed to the PCE of u ∗ . The variance σ 2 can be
decomposed into parameters that stress the contribution of each variable to the QoI,
in this case u ∗ . For instance, consider the stochastic space illustrated in Table 1.1.
Here, we consider a two-dimensional stochastic space, and we decompose the
variance of the solution as follows:
J. F. Reis et al.
Algorithm 3 Stochastic Galerkin algorithm to compute the PCE coefficients
1 Compute the coefficients k s (x) of the expansion in Eq. (1.26).
2 Compute matrices M s for each k s (x).
3 Compute the tensor C in Eq. (1.27).
4 Compute the block matrices in Eq. (1.28).
5 Solve the linear system of Eq. (1.29).
One way of computing the coefficients k s (x) is by using the KL expansion of
k(x, ξ ). This means we need to write the expansion in Eq. (1.13) in the form of
Eq. (1.26), and then we can use these coefficients to build the matrices M s . However,
there are ways of computing these matrices directly from the KL expansion of
k(x, ξ ); see [9]. Next, we need to compute the third order tensor C that has
(P G + 1) 3 entries. We can do so by exploiting the orthogonality properties of
the polynomials Ψ s (x). This yields a symmetric and sparse tensor that makes the
procedure much more efficient. Finally, after completing step 4 we solve the linear
system in Eq. (1.29). This is a sparse system of size N el (P G + 1) × N el (P G + 1).
Again, there are a number of techniques one could exploit to solve this system
in a more efficient way. These include the use of Krylov-based methods and preconditioning techniques [9].
1.6.4 Application of Surrogate Models: A Sensitivity Analysis
Using PC Expansions
We can exploit the coefficients of the PCE of a model to get statistical information
about the QoI, and this is a great improvement in terms of efficiency, if we were
to use a MC approach, instead. In this section, we discuss the computation of the
Sobol indices, a set of parameter which is very useful for sensitivity analysis. We
do not provide a detailed explanation about how to compute Sobol indices, the
interested reader can refer to [15] for details on this. The goal is to present a general
overview about differences in computing the Sobol indices using a MC approach
or a surrogate model approach. We also compute the Sobol indices for a specific
example, using the surrogate model approach, as in [9, 16].
Let ξ := {(ξ 1 , . . . , ξ N } be a sample of the germ, and consider the solution value
at the midpoint of the beam for this particular sample, i.e. u ∗ = u(x = 1/2, ξ). We
are interested in the significance of each random variable ξ i alone with respect to
the value u ∗ . This information is provided by the first order Sobol indices.
Let σ 2 be the variance of u ∗ . This variance accounts for the variability of all
random variables ξ i that contributed to the PCE of u ∗ . The variance σ 2 can be
decomposed into parameters that stress the contribution of each variable to the QoI,
in this case u ∗ . For instance, consider the stochastic space illustrated in Table 1.1.
Here, we consider a two-dimensional stochastic space, and we decompose the
variance of the solution as follows: