12
J. F. Reis et al.
a representation of a stochastic model based on the spectral decomposition of its
correlation function.
In general, it is possible to retrieve a linear operator K which is referred to as
the correlation kernel. We will avoid diving into the mathematical details of this
topic, the interested reader may refer to [9] to get a thorough and comprehensive
presentation of the subject. What is relevant here is that K owns some very useful
properties that allow us to retrieve a spectral expansion of the stochastic process U .
In particular, given its properties K has real non-negative eigenvalues λ i . For
each λ i there exist a finite number of linearly independent eigenvectors u i (x). The
collection of these eigenvectors constitutes the orthogonal basis upon which the KL
expansion is built. Therefore, it is possible to decompose the kernel as follow:
K(x, x
) =
i≥1
λ i u i (x)u i (x)
(1.11)
This is a common eigenproblem that can be solved using well-established
approaches.
Once the eigenvalues and their related eigenvectors are known, it is possible to
retrieve the KL expansion of the stochastic problem U which reads
U(x, θ) =
i≥1
λ i u i (x)η i (θ )
(1.12)
It is worth noting that the random variables η i (θ ) that enter in Eq. (1.12) have zero
mean, unit variance, and they are mutually uncorrelated.
This latter point is of the utmost importance. Indeed, if a certain stochastic
process depends on a given set of mutually correlated variables, one could exploit
the KL expansion to get a parametrisation of the stochastic space.
In particular, if the stochastic space includes a random field, the KL expansion
contains an infinite number of terms. In practice, we truncate the expansion, trading
some accuracy in place of a reduction of the dimensions of the stochastic space (and
a reduction of the computational effort). The truncated series in Eq. (1.13) counts a
number of terms at most equal to dimensions of the stochastic space, which in some
case could be very large.
ˆ
U(x, θ) :=
N KL
i≥1
λ i u i (x)η i (θ )
(1.13)
In order to set the truncation order N KL properly, one should make sure to include
the contribution of all the significant eigenmodes associated with the correlation
kernel. An important aspect that must be taken into account is that the rate of
decay of the eigenvalues is strictly related to the correlation length. Figure 1.3
reports the eigenvalues, normalised w.r.t. the largest one, as resulting from the
KL decomposition for different values of the correlation length l. The smaller
J. F. Reis et al.
a representation of a stochastic model based on the spectral decomposition of its
correlation function.
In general, it is possible to retrieve a linear operator K which is referred to as
the correlation kernel. We will avoid diving into the mathematical details of this
topic, the interested reader may refer to [9] to get a thorough and comprehensive
presentation of the subject. What is relevant here is that K owns some very useful
properties that allow us to retrieve a spectral expansion of the stochastic process U .
In particular, given its properties K has real non-negative eigenvalues λ i . For
each λ i there exist a finite number of linearly independent eigenvectors u i (x). The
collection of these eigenvectors constitutes the orthogonal basis upon which the KL
expansion is built. Therefore, it is possible to decompose the kernel as follow:
K(x, x
) =
i≥1
λ i u i (x)u i (x)
(1.11)
This is a common eigenproblem that can be solved using well-established
approaches.
Once the eigenvalues and their related eigenvectors are known, it is possible to
retrieve the KL expansion of the stochastic problem U which reads
U(x, θ) =
i≥1
λ i u i (x)η i (θ )
(1.12)
It is worth noting that the random variables η i (θ ) that enter in Eq. (1.12) have zero
mean, unit variance, and they are mutually uncorrelated.
This latter point is of the utmost importance. Indeed, if a certain stochastic
process depends on a given set of mutually correlated variables, one could exploit
the KL expansion to get a parametrisation of the stochastic space.
In particular, if the stochastic space includes a random field, the KL expansion
contains an infinite number of terms. In practice, we truncate the expansion, trading
some accuracy in place of a reduction of the dimensions of the stochastic space (and
a reduction of the computational effort). The truncated series in Eq. (1.13) counts a
number of terms at most equal to dimensions of the stochastic space, which in some
case could be very large.
ˆ
U(x, θ) :=
N KL
i≥1
λ i u i (x)η i (θ )
(1.13)
In order to set the truncation order N KL properly, one should make sure to include
the contribution of all the significant eigenmodes associated with the correlation
kernel. An important aspect that must be taken into account is that the rate of
decay of the eigenvalues is strictly related to the correlation length. Figure 1.3
reports the eigenvalues, normalised w.r.t. the largest one, as resulting from the
KL decomposition for different values of the correlation length l. The smaller