In this section, we describe more concretely how we determine the coefficients for the polynomial chaos expansions (PCE) described above. The expansion coefficients for the inputs are calculated using the characteristic information of the distribution of the input random variables. The coefficients for the probability density distribution of the response surface and the error of approximation are obtained using a probabilistic collocation approach that takes advantage of the form of the polynomial chaos expansions to minimize the number of model simulations required for a given level of accuracy.