On the use of the Sparse Grid techniques coupled with Polynomial Chaos

In this work we want to explore potentialities and deficiencies of Sparse Grid techniques coupled with Polynomial Chaos for multi dimensional (up to fifteen) stochastic problems. We used the sparse grid technique to compute the multi dimensional integrals needed to evaluate the coefficients of the polynomial expansion. Aim of this work is to compare several Sparse Grid techniques in terms of computational cost and accuracy with respect to Monte Carlo reference solution. We considered two problems: an algebraic function widely used in literature to test stochastic numerical methods, namely g-function, with poor regularity properties and a stochastic numerical simulation of a monodimensional compressible nozzle, where geometry and operating conditions are functions of random variables. After a detailed study on error computations and on the influence of the probability density function, we investigated the possibility of reducing the number of random variables by means of ANOVA analysis.