Spline Approximation on Sparse Grids

We present a compact sparse grid operator to multivariate functions on \([0,1]^d\) using the combination technique and tensor product spline interpolation. The construction is based on univariate spline interpolation using full end knots supported on \([0,1]\) , and the not-a-knot end condition. In this paper, we provide details for the construction of sparse grids and our spline interpolants, and a demonstration of computational performance. The paper includes a formula for counting the grid points in sparse grids, and an estimate for the number of computations in computing sparse spline interpolants. The results show that our construction performs at the level we expect based on other methods in the literature.