English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We consider the problem of approximating $[0,1]^{d}$-periodic functions byconvolution with a scaled Gaussian kernel. We start by establishing convergencerates to functions from periodic Sobolev spaces and we show that the saturationrate is $O(h^{2}),$ where $h$ is the scale of the Gaussian kernel. Taken from adiscrete point of view, this result can be interpreted as the accuracy that canbe achieved on the uniform grid with spacing $h.$ In the discrete setting, thecurse of dimensionality would place severe restrictions on the computation ofthe approximation. For instance, a spacing of $2^{-n}$ would provide anapproximation converging at a rate of $O(2^{-2n})$ but would require$(2^{n}+1)^{d}$ grid points. To overcome this we introduce a sparse gridversion of Gaussian convolution approximation, where substantially fewer gridpoints 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 thesparse grid setting (where the full grid error only deteriorates by a factor oforder $n^{d-1}$) however the analysis that leads to the result is novel in thatit draws on results from the theory of special functions and key observationsregarding the form of certain weighted geometric sums.

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