1 Introduction to Spectral Methods for Uncertainty Quantification
25
setting, the terms in the spectral sum are a pair or eigenvalues and eigenvectors,
where the latter depend only on the spatial coordinates. In the PCE case, there is a
third term that depends on the probabilistic space. This term is a polynomial, and it is
part of a basis of the probability space. Secondly, the coefficients are computed using
a Galerkin projection: the residuals are computed and projected onto the orthogonal
basis {Ψ s }. This procedure yields to the construction of a linear system which must
be solved to retrieve the deterministic coefficients of the PCE.
As mentioned, the stochastic Galerkin method is based on a reformulation of the
problem. Therefore, the approach requires full access to the equations. It is worth to
point out that this is not always granted, and, in some cases, it is even not feasible,
for instance, when we are dealing with a CFD framework. In this latter case, the
model is of the utmost complexity, since it is made up by algorithms, conditional
jumps, specific implementation choices for specific problems and so on. Therefore,
the possibility of exploiting a Galerkin approach for complex problems is often
highly questionable.
In the following, we will focus on our case study, the heat diffusion problem, for
which the Galerkin method is surely a suitable approach.
1.6.3.1 Weak Formulation and Deterministic Discretisation
Recall the mathematical formulation presented in Sect. 1.4.2. The goal of this
subsection is to present the weak formulation and the corresponding finite element
representation for both the stochastic and the deterministic problems. We rewrite the
deterministic model in Eq. (1.3),
∂ x k(x)∂ x u(x) = −f x ∈ (0, 1)
u(0) = u(1) = 0.
(1.20)
The weak formulation [2, 5] reads,
a(u, v) = b(v) for any v ∈ H
0
1 (0, 1)
(1.21)
where
a(u, v) =
1
0
k(x)∇u(x)∇v(x) dx and b(v) =
1
0
f (x)∇v(x) dx
The weak form is given by
A(u, v) = B(v), for any v ∈ H
0
1 (0, 1) ⊗ L
2 (Θ)
(1.22)
where
25
setting, the terms in the spectral sum are a pair or eigenvalues and eigenvectors,
where the latter depend only on the spatial coordinates. In the PCE case, there is a
third term that depends on the probabilistic space. This term is a polynomial, and it is
part of a basis of the probability space. Secondly, the coefficients are computed using
a Galerkin projection: the residuals are computed and projected onto the orthogonal
basis {Ψ s }. This procedure yields to the construction of a linear system which must
be solved to retrieve the deterministic coefficients of the PCE.
As mentioned, the stochastic Galerkin method is based on a reformulation of the
problem. Therefore, the approach requires full access to the equations. It is worth to
point out that this is not always granted, and, in some cases, it is even not feasible,
for instance, when we are dealing with a CFD framework. In this latter case, the
model is of the utmost complexity, since it is made up by algorithms, conditional
jumps, specific implementation choices for specific problems and so on. Therefore,
the possibility of exploiting a Galerkin approach for complex problems is often
highly questionable.
In the following, we will focus on our case study, the heat diffusion problem, for
which the Galerkin method is surely a suitable approach.
1.6.3.1 Weak Formulation and Deterministic Discretisation
Recall the mathematical formulation presented in Sect. 1.4.2. The goal of this
subsection is to present the weak formulation and the corresponding finite element
representation for both the stochastic and the deterministic problems. We rewrite the
deterministic model in Eq. (1.3),
∂ x k(x)∂ x u(x) = −f x ∈ (0, 1)
u(0) = u(1) = 0.
(1.20)
The weak formulation [2, 5] reads,
a(u, v) = b(v) for any v ∈ H
0
1 (0, 1)
(1.21)
where
a(u, v) =
1
0
k(x)∇u(x)∇v(x) dx and b(v) =
1
0
f (x)∇v(x) dx
The weak form is given by
A(u, v) = B(v), for any v ∈ H
0
1 (0, 1) ⊗ L
2 (Θ)
(1.22)
where