  A fast multilevel algorithm based on directionally scaled tensor-product
Gaussian kernels on structured sparse grids is proposed for interpolation of
high-dimensional functions and for the numerical integration of
high-dimensional integrals. The algorithm is based on the recent Multilevel
Sparse Kernel-based Interpolation (MLSKI) method (Georgoulis, Levesley \&
Subhan, \emph{SIAM J. Sci. Comput.}, 35(2), pp.~A815--A831, 2013), with
particular focus on the fast implementation of Gaussian-based MLSKI for
interpolation and integration problems of high-dimen-sional functions
$f:[0,1]^d\to\mathbb{R}$, with $5\le d\le 10$. The MLSKI interpolation
procedure is shown to be interpolatory and a fast implementation is proposed.
More specifically, exploiting the tensor-product nature of anisotropic Gaussian
kernels, one-dimensional cardinal basis functions on a sequence of hierarchical
equidistant nodes are precomputed to machine precision, rendering the
interpolation problem into a fully parallelisable ensemble of linear
combinations of function evaluations. A numerical integration algorithm is also
proposed, based on interpolating the (high-dimensional) integrand. A series of
numerical experiments highlights the applicability of the proposed algorithm
for interpolation and integration for up to 10-dimensional problems.
