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In this paper we present a locally and dimension-adaptive sparse grid methodfor interpolation and integration of high-dimensional functions withdiscontinuities. The proposed algorithm combines the strengths of thegeneralised sparse grid algorithm and hierarchical surplus-guided localadaptivity. A high-degree basis is used to obtain a high-order method which,given sufficient smoothness, performs significantly better than thepiecewise-linear basis. The underlying generalised sparse grid algorithmgreedily selects the dimensions and variable interactions that contribute mostto the variability of a function. The hierarchical surplus of points within thesparse grid is used as an error criterion for local refinement with the aim ofconcentrating computational effort within rapidly varying or discontinuousregions. This approach limits the number of points that are invested in`unimportant' dimensions and regions within the high-dimensional domain. Weshow the utility of the proposed method for non-smooth functions with hundredsof variables.

Local and Dimension Adaptive Sparse Grid Interpolation and Quadrature