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    Convergence of sparse grid Gaussian convolution approximation for multidimensional periodic functions

    Hubbert, Simon and Jaeger, J. and Levesley, J. (2023) Convergence of sparse grid Gaussian convolution approximation for multidimensional periodic functions. Applied and Computational Harmonic Analysis 62 , pp. 453-474. ISSN 1063-5203.

    SparseGaussianConvolution_REVISED2.pdf - Author's Accepted Manuscript
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    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 $\mathcal{O}(h^{2}),$ 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 $\mathcal{O}(2^{-2n})$ 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 (from $\mathcal{O}(2^{nd})$ on the full grid to just $\mathcal{O}(2^{n} n^{d-1})$ on the sparse grid) 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.


    Item Type: Article
    School: Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School
    Depositing User: Simon Hubbert
    Date Deposited: 31 Oct 2022 16:30
    Last Modified: 27 Oct 2023 00:10


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