Quasi-interpolation on a sparse grid with Gaussian

TL;DR

This paper introduces Q-MuSIK, a quasi-multilevel sparse kernel interpolation method that improves convergence and efficiency over traditional approaches, especially in high-dimensional problems, and develops a fast quadrature formula based on it.

Contribution

The paper presents a novel Q-MuSIK scheme that outperforms classical methods in convergence and runtime, and proposes a high-dimensional quadrature formula using this interpolation.

Findings

Q-MuSIK achieves better convergence than classical quasi-interpolation.

Q-MuSIK is more efficient in high-dimensional problems, avoiding large algebraic systems.

Numerical results demonstrate effectiveness in high-dimensional interpolation and quadrature.

Abstract

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.