Functional Chaos Expansion¶

Accounting for the joint probability density function (PDF)
 of the input random vector
, one seeks the joint PDF of the model response
. This may be achieved using
Monte Carlo (MC) simulation, i.e. by evaluating the model 
at a large number of realizations  of
 and then by estimating the empirical
distribution of the corresponding sample of model output
. However it is well-known that the MC
method requires a large number of model evaluations, i.e. a great
computational cost, in order to obtain accurate results.
In fact, when using MC simulation, each model run is performed
independently. Thus, whereas it is expected that
 if
, the model is
evaluated twice without accounting for this information. In other
words, the functional dependence between  and
 is lost.
A possible solution to overcome this problem and thereby to reduce the
computational cost of MC simulation is to represent the random
response  in a suitable functional space, such as
the Hilbert space  of square-integrable functions with
respect to the PDF .
Precisely, an expansion of the model response onto an orthonormal
basis of  is of interest.

The principles of the building of a (infinite numerable) basis of this
space, i.e. the PC basis, are described in the sequel.
Principle of the functional chaos expansion
Consider a model  depending on a set of random variables
. We call
functional chaos expansion the class of spectral methods which gathers
all types of response surface that can be seen as a projection of the
physical model in an orthonormal basis. This class of methods uses the
Hilbertian space (square-integrable space: ) to construct
the response surface.
Assuming that the physical model has a finite second order measure
(i.e. ), it may
be uniquely represented as a converging series onto an orthonormal
basis as follows:

$h(\underline{x})= \sum_{i=0}^{+\infty} \underline{y}_{i}\Phi_{i}(\underline{x}).$

where the $\underline{y}_{i} = (y_{i,1},\dots,y_{i,n_Y})^{\textsf{T}}$ ’s are deterministic vectors that fully characterize the random vector $\underline{Y}$ , and the $\Phi_{i}$ ’s are given basis functions (e.g. orthonormal polynomials, wavelets).

The orthonormality property of the functional chaos basis reads:

$\langle \Phi_{i},\Phi_{j}\rangle = \int_{D}\Phi_{i}(\underline{x}) \Phi_{j}(\underline{x})~f_{\underline{X}}(\underline{x}) d \underline{x} = \delta_{i,j}.$

where $\delta_{i,j} =1$ if $i=j$ and 0 otherwise. The metamodel $\widehat{h}(\underline{x})$ is represented by a finite subset of coefficients $\{y_{i}, i \in \cA \subset (N)\}$ in a truncated basis $\{\Phi_{i}, i \in \cA \subset (N)\}$ as follows:

$\widehat{h}(\underline{x})= \sum_{i \in \cA \subset N} y_{i}\Phi_{i}(\underline{x})$