Sparse Grid Approximation with Gaussians

Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in [Georgoulis, Levesley and Subhan, SIAM J. Sci. Comput., 35(2), pp. A815-A831, 2013], we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time in comparison with classical quasi-interpolation; namely, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional quadrature formula based on Q-MuSIK interpolation of the integrand. We present the results of numerical experimentation for both interpolation and quadrature in Rd, for d = 2, d = 3 and d = 4. In this work we also consider the convergence rates for multilevel quasiinterpolation of periodic functions using Gaussians on a grid. Initially, we have given the single level quasi-interpolation error by using the shifting properties of Gaussian kernel, and have then found an estimate for the multilevel error using the multilevel algorithm for unit function.