Convergence of sparse grid Gaussian convolution approximation for multi-dimensional periodic function

TL;DR

This paper develops a sparse grid Gaussian convolution method for approximating multi-dimensional periodic functions, achieving near-optimal convergence rates while significantly reducing computational complexity compared to full grid approaches.

Contribution

It introduces a novel sparse grid Gaussian convolution approximation method with proven convergence rates, overcoming the curse of dimensionality in high-dimensional settings.

Findings

Sparse grid approximation achieves $O(n^{d-1}2^{-2n})$ saturation rate.

Full grid approximation requires $(2^{n}+1)^{d}$ points for similar accuracy.

The analysis combines special functions theory and geometric sums to establish convergence.

Abstract

We consider the problem of approximating $[0,1]^{d}$-periodic functions by convolution with a scaled Gaussian kernel. We start by establishing convergence rates to functions from periodic Sobolev spaces and we show that the saturation rate is $O(h^{2}),$ where $h$ is the scale of the Gaussian kernel. Taken from a discrete point of view, this result can be interpreted as the accuracy that can be achieved on the uniform grid with spacing $h.$ In the discrete setting, the curse of dimensionality would place severe restrictions on the computation of the approximation. For instance, a spacing of $2^{-n}$ would provide an approximation converging at a rate of $O(2^{-2n})$ but would require $(2^{n}+1)^{d}$ grid points. To overcome this we introduce a sparse grid version of Gaussian convolution approximation, where substantially fewer grid points are required, and show that the sparse grid version delivers a saturation rate of $O(n^{d-1}2^{-2n}).$ This rate is in line with what one would expect in the sparse grid setting (where the full grid error only deteriorates by a factor of order $n^{d-1}$) however the analysis that leads to the result is novel in that it draws on results from the theory of special functions and key observations regarding the form of certain weighted geometric sums.