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 finite subset of coefficients in a truncated basis as follows:
As an example of this type of expansion, one can mention responses by wavelet expansion, polynomial chaos expansion, etc.