Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation

Abstract

Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Matérn kernels ${\Phi }_{\mathbf{\nu },\mathbf{\lambda }}(\mathbf{x},{\mathbf{x}}^{\prime })={\prod }_{j=1}^d{\varphi }_{{\nu }_j,{\lambda }_j}(x_j,x_j^{\prime })$, with a particular focus on the anisotropic setting where the regularity ${\nu }_j$ and the lengthscale ${\lambda }_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic ${\nu }_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic ${\lambda }_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation.

16 pages, 4 figures