Hierarchical Extended B-splines for Approximations on Sparse Grids

B-splines on sparse grids are a versatile method for higher-dimensional approximation, especially in smooth settings or if gradient information is advantageous. With increasing dimension, grid points on the boundary become prohibitively expensive and have to be avoided. To compensate for the missing boundary points, the basis functions are typically modified according to natural boundary conditions. However, the resulting splines do not span the polynomial space anymore. In this work we introduce hierarchical extended B-splines that are based on the polynomial basis property. We apply them on spatially adaptive sparse grids and show improved convergence results for analytical objective functions and a real-world subsurface flow model.