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

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 ${\mathrm{\Phi }}_{\mathit{\nu },\mathit{\lambda }}(\mathbf{x},{\mathbf{x}}^{\mathrm{\prime }})={\prod }_{j=1}^d{\phi }_{{\nu }_j,{\lambda }_j}(x_j,x_j^{\mathrm{\prime }})$, 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.