Multi-element stochastic spectral projection for high quantile estimation

We investigate quantile estimation by multi–element generalized Polynomial Chaos (gPC) metamodel where the exact numerical model is approximated by complementary metamodels in overlapping domains that mimic the model’s exact response. The gPC metamodel is constructed by the non–intrusive stochastic spectral projection approach and function evaluation on the gPC metamodel can be considered as essentially free. Thus, large number of Monte Carlo samples from the metamodel can be used to estimate ↵–quantile, for moderate values of ↵. As the gPC metamodel is an expansion about the means of the inputs, its accuracy may worsen away from these mean values where the extreme events may occur. By increasing the approximation accuracy of the metamodel, we may eventually improve accuracy of quantile estimation but it is very expensive. A multi–element approach is therefore proposed by combining a global metamodel in the standard normal space with supplementary local metamodels constructed in a bounded domain about the design points corresponding to the extreme events. To improve the accuracy and to minimize the sampling cost, sparse–tensor and anisotropic–tensor quadrature are tested in addition to the full–tensor Gauss quadrature in the construction of local metamodels; di↵erent bounds of the gPC expansion are also examined. The global and local metamodels are combined in the multi–element gPC (MEgPC) approach and it is shown that MEgPC can be more accurate than Monte Carlo or importance sampling methods for high quantile estimations for input dimensions roughly below N = 8, a limit that is very much case– and ↵–dependent.