The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matern covariances with smoothness index $\nu$ and correlation length $\lambda$, we analyse the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.

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Matérn 协方差的基于周期的采样方法的统一分析

在包含感兴趣空间域的足够大的圆环上对固定高斯随机场进行周期化是各种有效计算方法的基础，例如使用快速傅立叶变换在均匀网格上生成样本的经典循环嵌入技术。对于具有平滑指数 $\nu$ 和相关长度 $\lambda$ 的 Matern 协方差族，我们分析了非平滑周期化（对应于经典循环嵌入）和使用协方差函数的平滑截断的替代程序。我们解决了两个开放问题：第一个关于非光滑情况下所得循环的特征值的 $\nu$ 依赖渐近衰减，第二个关于环面的 $\nu$、$\lambda$ 所需的大小使用平滑周期时。在此过程中，我们对这两种方法的性能进行了完整的表征。我们的理论估计和此处提供的数值测试都显示了平滑截断的巨大优势。