Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation
Provides theoretical and numerical evidence for improved sparse grid interpolation in high-dimensional anisotropic function approximation.
This work studies sparse grid interpolation with separable Matérn kernels in anisotropic settings, showing that combining anisotropic regularity and lengthscale information improves both asymptotic and pre-asymptotic convergence rates.
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 $Φ_{\boldsymbolν,\boldsymbolλ}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^dϕ_{ν_j,λ_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $ν_j$ and the lengthscale $λ_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $ν_j$ 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 $λ_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation.