Introduction to PC decomposition.
Assume that X is a multivariate random variable, with probability distribution function f. Consider the nonlinear function Y=g(X), where g is a square integrable nonlinear function.
The polynomial chaos decomposition of g is
where P is the number of multivariate chaos polynomials
and is the k-th multivariate chaos polynomial.
These multivariate polynomials are constructed from tensorisation of univariate
orthogonal polynomials.
The coefficients of the PC decomposition satisfy the equation:
The denominator of the previous equation can be computed in advance, since the associated integrals only depend on the chaos polynomials, which have known integrals. Therefore, only the numerator has to be computed. There are two different methods to do this: integration or regression.
In this section, we present the orthogonal polynomials which are used to create the model. For each distribution associated to the random variable, we use a different type of orthogonal polynomial. The chosen family of polynomials has the property that they are orthogonal with respect to the weight associated with the probability distribution function of the variable.
Distribution | Polynomial |
"Uniforme" | Legendre |
"LogUniforme" | Legendre |
"Normale" | Hermite |
"LogNormale" | Hermite |
"Exponentielle" | Laguerre |
Let us denote by a family of
univariate orthogonal polynomials.
The associated multivariate chaos polynomial is
where is the i-th multi-indice
associated with the polynomial
.
The two methods to compute the PC decomposition in this toolbox are integration, based on quadrature multidimensionnal rules, and regression, based on the solution of a linear least squares problem.
The integration method uses a multidimensionnal quadrature based on the formula
where n is the number of integration points,
are the integration weights and
are the integration points.
An integration rule, such as the "Quadrature"
sampling, for example, defines the number of integration points, the weights and
the integration points.
In the regression method, the coefficients of the PC decomposition are
computed by solving a linear least squares problem.
The coefficients minimize
Hence, the type of sampling used to create the polynomial chaos decomposition
must be consistent with the method used to compute the
coefficients of the decomposition.
The name
argument of the
setrandvar_buildsample
function
must be consistent with the method
argument of the polychaos_computeexp
function.
These two functions are generally combined, as in the following fragment
of script.
setrandvar_buildsample(srvx,name,np) [...] polychaos_computeexp(pc,srvx,method)
The next table presents the available combinations.
Sample name | "Regression" | "Integration" |
"MonteCarlo" | OK | |
"Lhs" | OK | |
"QmcSobol" | OK | |
"Quadrature" | OK | |
"Petras" | OK | |
"SmolyakGauss" | OK | |
"SmolyakFejer" | OK | |
"SmolyakTrapeze" | OK | |
"SmolyakClenshawCurtis" | OK |