Elliot J. Addy | Searcharxiv

arXiv2026

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 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.

math.NA↗

arXiv2025

Lengthscale-informed sparse grids for kernel methods in high dimensions

Kernel interpolation, especially in the context of Gaussian process emulation, is a widely used technique in surrogate modelling, where the goal is to cheaply approximate an input-output map using a limited number of function evaluations. However, in high-dimensional settings, such methods typically suffer from the curse of dimensionality; the number of required evaluations to achieve a fixed approximation error grows exponentially with the input dimension. To overcome this, a common technique used in high-dimensional approximation methods, such as quasi-Monte Carlo and sparse grids, is to exploit functional anisotropy: the idea that some input dimensions are more 'sensitive' than others. In doing so, such methods can significantly reduce the dimension dependence in the error. In this work, we propose a generalisation of sparse grid methods that incorporates a form of anisotropy encoded by the lengthscale parameter in Mat\'ern kernels. We derive error bounds and perform numerical experiments that show that our approach enables effective emulation over arbitrarily high dimensions for functions exhibiting sufficient anisotropy.

math.NA↗