  High-dimensional interpolation problems appear in various applications of
uncertainty quantification, stochastic optimization and machine learning. Such
problems are computationally expensive and request the use of adaptive grid
generation strategies like anisotropic sparse grids to mitigate the curse of
dimensionality. However, it is well known that the standard dimension-adaptive
sparse grid method converges very slowly or even fails in the case of
non-smooth functions. For piecewise smooth functions with kinks, we construct
two novel hp-adaptive sparse grid collocation algorithms that combine low-order
basis functions with local support in parts of the domain with less regularity
and variable-order basis functions elsewhere. Spatial refinement is realized by
means of a hierarchical multivariate knot tree which allows the construction of
localised hierarchical basis functions with varying order. Hierarchical surplus
is used as an error indicator to automatically detect the non-smooth region and
adaptively refine the collocation points there. The local polynomial degrees
are optionally selected by a greedy approach or a kink detection procedure.
Three numerical benchmark examples with different dimensions are discussed and
comparison with locally linear and highest degree basis functions are given to
show the efficiency and accuracy of the proposed methods.
