论文标题
多维周期功能的稀疏网格高斯卷积近似的收敛性
Convergence of sparse grid Gaussian convolution approximation for multi-dimensional periodic function
论文作者
论文摘要
我们考虑近似$ [0,1]^{d} $的问题 - 与缩放的高斯内核进行卷积的定期功能。首先，我们将收敛速率建立到定期Sobolev空间的功能，并表明饱和速率为$ O（H^{2}），其中$ h $是高斯内核的规模。从离散的角度来看，该结果可以解释为在离散设置的间距$ h。$ h。$ h。$h。$上可以实现的准确性，维度的诅咒将对近似值的计算施加严重限制。例如，$ 2^{ - n} $的间距将以$ O（2^{ - 2n}）$的速率提供近似收敛，但需要$（2^{n} +1）^{d} $ grid点。为了克服这一点 表明稀疏的网格版本提供$ O（n^{d-1} 2^{ - 2n}）的饱和速率。$此速率与一个人在稀疏网格设置中的期望一致（在稀疏的网格设置中，只有完整的网格误差仅通过$ n^{d-1} $的分析而导致的分析及其绘制的功能，并且绘制了该图的功能，并且绘制了该图的功能，并且该图是绘制的。加权几何总和。
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.
