In this paper we present a stochastic collocation method for quantifying uncertainty in models with large numbers of uncertain inputs and non-smooth input-output maps. The proposed algorithm combines the strengths of dimension adaptivity and hierarchical surplus guided local adaptivity to facilitate computationally efficient approximation of models with bifurcations/discontinuties in high-dimensional input spaces. A comparison is made against two existing stochastic collocation methods and found, in the cases tested, to significantly reduce the number of model evaluations needed to construct an accurate surrogate model. The proposed method is then used to quantify uncertainty in a model of flow through porous media with an unknown permeability field. A Karhunen-Loeve expansion is used to parameterize the uncertainty and the resulting mean and variance in the speed of the fluid and the time dependent saturation front are computed.
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