Lengthscale-informed sparse grids for kernel methods in high dimensions

Elliot J. Addy

doi:10.1007/s00211-026-01527-5

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Lengthscale-informed sparse grids for kernel methods in high dimensions

2026-02-01

Elliot J. Addy *

Jonas Latz

Aretha L. Teckentrup

DOI：10.1007/s00211-026-01527-5

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Abstract

En 中文

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 & eacute;rn 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.

Keywords:

DIFFERENTIAL-EQUATIONS

REGRESSION

APPROXIMATION

INTERPOLATION

CONVERGENCE

CALIBRATION

INTEGRATION

ALGORITHMS

BOUNDS

FIELD

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