Quasi-interpolation on a sparse grid with Gaussian
This work addresses the need for efficient high-dimensional interpolation and quadrature, offering a faster alternative to existing methods, though it is an incremental improvement over prior work.
The authors propose Q-MuSIK, a quasi-multilevel sparse interpolation method using kernels, which achieves better convergence and run time than classical quasi-interpolation, especially in high dimensions, by avoiding large algebraic systems. Numerical experiments demonstrate improvements in both interpolation and quadrature.
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 high dimension.