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

TL;DR

This paper analyzes the convergence properties of sparse grid kernel interpolation using anisotropic Matérn kernels, highlighting improvements in high-dimensional function approximation.

Contribution

It introduces a combined approach of anisotropic and lengthscale-informed sparse grids, with theoretical analysis and numerical validation.

Findings

Improved convergence rates in smooth dimensions.

Diminished error contribution from less varying dimensions.

Both asymptotic and pre-asymptotic error behaviors are characterized.

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 Mat\'ern kernels $\Phi_{\boldsymbol{\nu},\boldsymbol{\lambda}}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^d\phi_{\nu_j,\lambda_j}(x_j,x_j')$, 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.