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

被引:2
作者
Hubbert, Simon [1 ]
Jaeger, Janin [2 ]
Levesley, Jeremy [3 ]
机构
[1] Birkbeck Univ London, Dept Econ Math & Stat, London WC1H 7HX, England
[2] Justus Liebig Univ, Lehrstuhl Numer Math, D-35392 Giessen, Germany
[3] Univ Leicester, Dept Math, Leicester LE1 7RH, Leics, England
关键词
Sparse grids; Gaussian convolution; Approximation of periodic functions;
D O I
10.1016/j.acha.2022.10.005
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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(h2), 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 (2n + 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 O(2nd) on the full grid to just O(2nnd-1) on the sparse grid) and show that the sparse grid version delivers a saturation rate of O(nd-12-2n). 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 nd-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.(c) 2022 Elsevier Inc. All rights reserved.
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页码:453 / 474
页数:22
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