1 Introduction to Spectral Methods for Uncertainty Quantification
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given the altitude value and not told that it had been sampled in a plain region, we
could not have any idea of what the territory looks like.
It is needless to say that the very same reasoning applies to any stochastic field
that we may find in any kind of application.
1.4 Sampling Techniques
The evolution of a physical process depends on parameters, the inputs, which are
uncertain, and, in general, we are provided with their probability distributions in the
stochastic space.
When we are dealing with a set of independent variables, we can just sample
independently from each probability distribution, to reconstruct the vector of
stochastic inputs. On the other hand, when the random variables are dependent,
we should sample from their joint probability distribution. This is usually not
straightforward, and we generally need to exploit special techniques.
For a jointly distributed set of variables θ ∈ Θ, where Θ is a d-dimensional set,
it is possible to define a (d × d) correlation matrix where each entry corresponds to
the correlation coefficient among the i-th and the j -th variable.
One way to accomplish the sampling of multiple variables could be to decompose
the correlation matrix into the product of a lower triangular matrix L and its
conjugate transpose, according to the Cholesky method. Once L is available, it is
possible to retrieve the input random vector as
θ = Lη
(1.10)
where η is a (d × 1)-dimensional vector whose elements can be sampled independently from a normal distribution η ∼ N (0, 1).
An alternative way to proceed with the sampling of jointly distributed variables
is to rely on a parametrisation of the stochastic space. This means establishing a
functional relation that maps a set of independent random variables, hereinafter
referred to as the germ, to the initial mutually correlated parameters. Differently than
the Cholesky decomposition approach, which leaves the dimension of the stochastic
space unchanged, in some cases the parametrisation opens the path to an order
reduction of the problem. This is particularly useful when dealing with stochastic
fields.
1.4.1 Karhunen–Loève Expansion
The Karhunen–Loève decomposition (KL), or proper orthogonal decomposition
(POD), was first proposed in the 1940s by Kac and Siegert [6], Karhunen [7] and
Loeve [8]. The main idea upon which the KL expansion relies on is to provide
11
given the altitude value and not told that it had been sampled in a plain region, we
could not have any idea of what the territory looks like.
It is needless to say that the very same reasoning applies to any stochastic field
that we may find in any kind of application.
1.4 Sampling Techniques
The evolution of a physical process depends on parameters, the inputs, which are
uncertain, and, in general, we are provided with their probability distributions in the
stochastic space.
When we are dealing with a set of independent variables, we can just sample
independently from each probability distribution, to reconstruct the vector of
stochastic inputs. On the other hand, when the random variables are dependent,
we should sample from their joint probability distribution. This is usually not
straightforward, and we generally need to exploit special techniques.
For a jointly distributed set of variables θ ∈ Θ, where Θ is a d-dimensional set,
it is possible to define a (d × d) correlation matrix where each entry corresponds to
the correlation coefficient among the i-th and the j -th variable.
One way to accomplish the sampling of multiple variables could be to decompose
the correlation matrix into the product of a lower triangular matrix L and its
conjugate transpose, according to the Cholesky method. Once L is available, it is
possible to retrieve the input random vector as
θ = Lη
(1.10)
where η is a (d × 1)-dimensional vector whose elements can be sampled independently from a normal distribution η ∼ N (0, 1).
An alternative way to proceed with the sampling of jointly distributed variables
is to rely on a parametrisation of the stochastic space. This means establishing a
functional relation that maps a set of independent random variables, hereinafter
referred to as the germ, to the initial mutually correlated parameters. Differently than
the Cholesky decomposition approach, which leaves the dimension of the stochastic
space unchanged, in some cases the parametrisation opens the path to an order
reduction of the problem. This is particularly useful when dealing with stochastic
fields.
1.4.1 Karhunen–Loève Expansion
The Karhunen–Loève decomposition (KL), or proper orthogonal decomposition
(POD), was first proposed in the 1940s by Kac and Siegert [6], Karhunen [7] and
Loeve [8]. The main idea upon which the KL expansion relies on is to provide