Functional Chaos Expansion

Accounting for the joint probability density function (PDF) f_{\underline{X}}(\underline{x}) of the input random vector \underline{X}, one seeks the joint PDF of the model response \underline{Y} = h(\underline{X}). This may be achieved using Monte Carlo (MC) simulation, i.e. by evaluating the model h at a large number of realizations \underline{x}^{(i)} of \underline{X} and then by estimating the empirical distribution of the corresponding sample of model output h(\underline{x}^{(i)}). 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 h(\underline{x}^{(i)}) \approx h(\underline{x}^{(j)}) if \underline{x}^{(i)} \approx \underline{x}^{(j)}, the model is evaluated twice without accounting for this information. In other words, the functional dependence between \underline{X} and \underline{Y} 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 \underline{Y} in a suitable functional space, such as the Hilbert space L^2 of square-integrable functions with respect to the PDF f_{\underline{X}}(\underline{x}). Precisely, an expansion of the model response onto an orthonormal basis of L^2 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 h depending on a set of random variables \underline{X} = (X_1,\dots,X_{n_X})^{\textsf{T}}. 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: L^2) to construct the response surface.
Assuming that the physical model has a finite second order measure (i.e. E\left( \|h(\underline{X})\|^2\right)< + \infty), 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})

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