# 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:

where the ’s are deterministic vectors that fully characterize the random vector , and the ’s are given basis functions (e.g. orthonormal polynomials, wavelets).

The orthonormality property of the functional chaos basis reads:

where if and 0 otherwise. The metamodel is represented by a

finitesubset of coefficients in atruncatedbasis as follows:

As an example of this type of expansion, one can mention responses by wavelet expansion, polynomial chaos expansion, etc.