1 Introduction to Spectral Methods for Uncertainty Quantification
7
1.2.2 The Stochastic Heat Diffusion Equation
The heat diffusion problem may be affected by a number of uncertainties. For
instance, the temperature at one of the edges of the beam may be known within
a given interval of confidence, or the length of the beam itself may be uncertain.
To account for the uncertainty affecting a quantity, for instance, the temperature
imposed at a boundary, one should avoid using a deterministic value and assign a
probability density function to the variable.
In this section, we consider one source of uncertainty only: the one related to the
thermal conductivity term k. Under the mentioned hypotheses, the stochastic heat
diffusion model reads,
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ x k(x, θ)∂ x u(x, θ ) = −f x ∈ (0, 1), θ ∈ Θ.
u(0, θ) = u 0
u(1, θ) = u 1
(1.5)
Equation (1.5) is a generalisation of Eq. (1.3), where k(x, θ) is a random field or a
stochastic field. A random field is a map that receives two types of inputs: a random
variable, in this case θ , and second variable from a deterministic space, in this case
x. In other words, this means that k(x, θ) is a function that does not just depend
in space, but it also depends on the random element θ . This random element θ
could either be a variable or a vector, and it belongs to the set Θ. If θ is, say, a
normal distributed variable, then k(x, θ) is a surface in k(x, θ) : (0, 1) × R → R.
Usually, the complexity of the physics implies that Θ is a multidimensional set. The
dependency on θ is a choice we make a-priori, and the process to write k(x) w.r.t. θ
is called parametrisation. We will see later in this chapter how this parametrisation
is done and how useful it will be.
In general, we are concerned with the evaluation of a QoI resulting from the
solution of Eq. (1.5) rather than the solution itself. The QoI may be any statistical
value, like the mean and the variance, or any higher statistical momentum. In some
cases one could be interested in assessing the sensitivity of the solution with respect
to an input parameter.
Some of this information can be easily computed when k(x, θ) ≡ k(θ), i.e., k is
homogeneous; thus it does not depend in space. Nevertheless, as a function of two
variables, the fact that k is constant in x does not imply that it is a constant variable.
Indeed, it is still a stochastic space that depends on the random parameter θ . For
instance, using the solution in Eq. (1.4), we can readily compute the mean of higher
moments of u(x, θ ). For instance, if we consider a log-normal-distributed variable
k with mean μ and variance σ , the mean solution (Fig. 1.2) is given by,
E [u(x, k)] = −
f x 2
2E[k]
+
u 1 +
1
2E[k]
− u 0
x + u 0 ,
(1.6)
7
1.2.2 The Stochastic Heat Diffusion Equation
The heat diffusion problem may be affected by a number of uncertainties. For
instance, the temperature at one of the edges of the beam may be known within
a given interval of confidence, or the length of the beam itself may be uncertain.
To account for the uncertainty affecting a quantity, for instance, the temperature
imposed at a boundary, one should avoid using a deterministic value and assign a
probability density function to the variable.
In this section, we consider one source of uncertainty only: the one related to the
thermal conductivity term k. Under the mentioned hypotheses, the stochastic heat
diffusion model reads,
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
∂ x k(x, θ)∂ x u(x, θ ) = −f x ∈ (0, 1), θ ∈ Θ.
u(0, θ) = u 0
u(1, θ) = u 1
(1.5)
Equation (1.5) is a generalisation of Eq. (1.3), where k(x, θ) is a random field or a
stochastic field. A random field is a map that receives two types of inputs: a random
variable, in this case θ , and second variable from a deterministic space, in this case
x. In other words, this means that k(x, θ) is a function that does not just depend
in space, but it also depends on the random element θ . This random element θ
could either be a variable or a vector, and it belongs to the set Θ. If θ is, say, a
normal distributed variable, then k(x, θ) is a surface in k(x, θ) : (0, 1) × R → R.
Usually, the complexity of the physics implies that Θ is a multidimensional set. The
dependency on θ is a choice we make a-priori, and the process to write k(x) w.r.t. θ
is called parametrisation. We will see later in this chapter how this parametrisation
is done and how useful it will be.
In general, we are concerned with the evaluation of a QoI resulting from the
solution of Eq. (1.5) rather than the solution itself. The QoI may be any statistical
value, like the mean and the variance, or any higher statistical momentum. In some
cases one could be interested in assessing the sensitivity of the solution with respect
to an input parameter.
Some of this information can be easily computed when k(x, θ) ≡ k(θ), i.e., k is
homogeneous; thus it does not depend in space. Nevertheless, as a function of two
variables, the fact that k is constant in x does not imply that it is a constant variable.
Indeed, it is still a stochastic space that depends on the random parameter θ . For
instance, using the solution in Eq. (1.4), we can readily compute the mean of higher
moments of u(x, θ ). For instance, if we consider a log-normal-distributed variable
k with mean μ and variance σ , the mean solution (Fig. 1.2) is given by,
E [u(x, k)] = −
f x 2
2E[k]
+
u 1 +
1
2E[k]
− u 0
x + u 0 ,
(1.6)