1 Introduction to Spectral Methods for Uncertainty Quantification
15
Considering the space of stochastic processes,
L
2 (Ξ ) := {ϕ(ξ ) :
Ξ
ϕ(ξ )
2 dμ(ξ ) < ∞}
where
dμ(ξ ) = f Ξ (ξ )dξ
and f Ξ (ξ ) is the probability density function (PDF) of the multivariate standard
Gaussian random variable ξ . Now, because of independence the PDF f Ξ (ξ ) is
the product of each PDF f Ξ i (ξ i ) of each one-dimensional random variable ξ i ,
i.e., f Ξ (ξ ) =
N
i=1 f Ξ i (ξ i ). This would not happen, if the variable were not
independent. It is important to understand that this makes all the theory much
simpler and saves a great deal of computational effort. Indeed, it is one of many
benefits of choosing a parametrisation, such as the KL expansion for normally
distributed stochastic processes.
Remark 1.1 The notation dμ(ξ ) represents the measure of the probability space. If
the measure of Ξ is the one-dimensional standard Gaussian measure then,
Ξ
ϕ(ξ )dμ(ξ ) =
∞
−∞
ϕ(ξ )
1
√
2π
e
−ξ 2 /2 dξ
Moreover, considering the space,
H
1
0 (0, 1) :=
φ(x) :
1
0
φ
2 dx < ∞ and
1
0
(φ
)
2 < ∞ and φ(0) = φ(1) = 0
which is a Sobolev space. Spaces, such as H 1
0 (0, 1), are important, because a
function φ ∈ H 1
0 (0, 1) has peculiar properties. For instance, the Lax–Milgram
theorem [5], which gives necessary and sufficient conditions for existence and
uniqueness of solution for Eq. (1.5), is valid. These results are the backbone of UQ
theory, and they should be kept in mind. In the following, superscript 1 implies that
the first derivative of the function is squared and integrable, while the subscript 0
points out that the function vanishes at the boundary. The (0, 1) refers to the domain
in which the function is defined. Finally, we denote the {φ i } ∞
i=1 an orthogonal basis
of the space H 1
0 (0, 1).
The solution of Eq. (1.15) is a stochastic process u(x, ξ ) ∈ H 1
0 (0, 1) ⊗ L 2 (Ξ ).
This means that if we focus on the deterministic part of u, i.e., if we freeze the
variable θ and let x free to vary, then u(x, ·) ∈ H 1
0 (0, 1). If we do the opposite and
focus on the probabilistic part of u, then u(·, ξ) ∈ L 2 (Ξ ).
In particular, if we consider the deterministic heat diffusion equation (1.1), the
solution does not have any probabilistic component; therefore u(x) ∈ H 1
0 (0, 1).
15
Considering the space of stochastic processes,
L
2 (Ξ ) := {ϕ(ξ ) :
Ξ
ϕ(ξ )
2 dμ(ξ ) < ∞}
where
dμ(ξ ) = f Ξ (ξ )dξ
and f Ξ (ξ ) is the probability density function (PDF) of the multivariate standard
Gaussian random variable ξ . Now, because of independence the PDF f Ξ (ξ ) is
the product of each PDF f Ξ i (ξ i ) of each one-dimensional random variable ξ i ,
i.e., f Ξ (ξ ) =
N
i=1 f Ξ i (ξ i ). This would not happen, if the variable were not
independent. It is important to understand that this makes all the theory much
simpler and saves a great deal of computational effort. Indeed, it is one of many
benefits of choosing a parametrisation, such as the KL expansion for normally
distributed stochastic processes.
Remark 1.1 The notation dμ(ξ ) represents the measure of the probability space. If
the measure of Ξ is the one-dimensional standard Gaussian measure then,
Ξ
ϕ(ξ )dμ(ξ ) =
∞
−∞
ϕ(ξ )
1
√
2π
e
−ξ 2 /2 dξ
Moreover, considering the space,
H
1
0 (0, 1) :=
φ(x) :
1
0
φ
2 dx < ∞ and
1
0
(φ
)
2 < ∞ and φ(0) = φ(1) = 0
which is a Sobolev space. Spaces, such as H 1
0 (0, 1), are important, because a
function φ ∈ H 1
0 (0, 1) has peculiar properties. For instance, the Lax–Milgram
theorem [5], which gives necessary and sufficient conditions for existence and
uniqueness of solution for Eq. (1.5), is valid. These results are the backbone of UQ
theory, and they should be kept in mind. In the following, superscript 1 implies that
the first derivative of the function is squared and integrable, while the subscript 0
points out that the function vanishes at the boundary. The (0, 1) refers to the domain
in which the function is defined. Finally, we denote the {φ i } ∞
i=1 an orthogonal basis
of the space H 1
0 (0, 1).
The solution of Eq. (1.15) is a stochastic process u(x, ξ ) ∈ H 1
0 (0, 1) ⊗ L 2 (Ξ ).
This means that if we focus on the deterministic part of u, i.e., if we freeze the
variable θ and let x free to vary, then u(x, ·) ∈ H 1
0 (0, 1). If we do the opposite and
focus on the probabilistic part of u, then u(·, ξ) ∈ L 2 (Ξ ).
In particular, if we consider the deterministic heat diffusion equation (1.1), the
solution does not have any probabilistic component; therefore u(x) ∈ H 1
0 (0, 1).