Hybrid Sampling for Uncertainty Quantification in Systems with High Dimensional Parameter Spaces

Abstract

Spatial models formulated as partial differential equations, often include space dependent parameters that are not readily estimable and are therefore uncertain. There are two broad classes of methods for computing statistical quantities of inter est related to the model solution: Spectral methods, such as the generalized polyno mial chaos and stochastic collocation methods, are well-suited for systems with low parameter complexity. However, their convergence rates deteriorate as the dimen sion of the parameter space increases, and hence for systems with high parameter complexity, methods whose convergence rates are independent of the stochastic di mension, such as the Monte Carlo method, are more appropriate. In this work, we propose a hybrid sampling scheme which uses conditional sampling to combine sparse grid quadrature rules on a low-dimensional projection of the parameter space with a Monte Carlo scheme to compensate in the remaining dimensions. Using complexity arguments, we show that our method is more efficient than either of its constituents. We include some numerical examples to illustrate our results.