# GPCE.quad: Solve polynomial chaos expansion with numerical quadrature In GPC: Generalized Polynomial Chaos

## Description

The GPCE.quad function implements a polynomial chaos expansion of a given model or an external model. The strategy for the expansion of the model into a polynomial chaos basis is the Gauss quadrature method where the Gauss quadrature points are used to estimate the integrales corresponding to the coefficients of the expansion. A statistical and a global sensitivity analysis of the model is then carried out.

## Arguments

 InputDim Number of input random variables PCSpace To decide if it should be removed after discussion with Miguel.. InputDistrib Distribution of the input. Options are Gaussian, Uniform, Beta, Gamma ParamDistrib Shape parameters of the random variables. Model A function defining the model to analyze or NULL if the model is external ModelParam Optional parameters for function evaluation. Output The results of the model evaluation at the quadrature points. DesignInput To decide if it should be removed after discussion with Miguel. p The order of the polynomial chaos expansion.

order)

 ExpPoly The polynomial used in the expansion. Options are Hermite, Legendre, Jacobi, Laguerre QuadType Type of quadrature. Options are SPARSE or FULL QuadPoly The type of one-dimensional quadrature rule to be used. For SPARSE, one can use ClenshawCurtis or Fejer. For FULL, one could choose Hermite,Legendre,Jacobi or Laguerre QuadLevel Level of quadrature used in the approximation.

## Value

 Designs A list containing the quadrature size, n, a vector of the n quadrature weights, and a n * d matrix of the quadrature points Output Vector of the model output PCEcoeff Matrix of the kept sparse polynomial basis. TruncSet_ij is the jth polynomial degree associated to the ith variable Moments A list containing the fourth first moments of the output: mean, variance, standard deviation, skweness and kurtosis Sensitivity A list containing the sobol sensitivity indices (Values) and the normalized Sobol nominal and total sensitivity indices (S and ST)

Jordan Ko

## References

J. Ko, D. Lucor and P. Sagaut, 2008, On Sensitivity of Two-Dimensional Spatially Developing Mixing Layer With Respect to Uncertain Inflow Conditions, Physics of Fluids, 20(7), 07710201-07710220.

J. Ko, 2009, Applications of the generalized polynomial chaos to the numerical simulationof stochastic shear flows, Doctoral thesis, University of Paris VI.

J. Ko, D. Lucor and P. Sagaut, 2011, Effects of base flow uncertainty on Couette flow stability, Computers and Fluids, 43(1), 82-89.