Gaussian processes (GPs) and Gaussian random fields (GRFs) are essential for modelling spatially varying stochastic phenomena. Yet, the efficient generation of corresponding realisations on high-resolution grids remains challenging, particularly when a large number of realisations are required. This paper presents two novel contributions. First, we propose a new methodology based on Dirichlet-Neumann averaging (DNA) to generate GPs and GRFs with isotropic covariance on regularly spaced grids. The combination of discrete cosine and sine transforms in the DNA sampling approach allows for rapid evaluations without the need for modification or padding of the desired covariance function. While this introduces an error in the covariance, our numerical experiments show that this error is negligible for most relevant applications, representing a trade-off between efficiency and precision. We provide explicit error estimates for Mat\'ern covariances. The second contribution links our new methodology to the stochastic partial differential equation (SPDE) approach for sampling GRFs. We demonstrate that the concepts developed in our methodology can also guide the selection of boundary conditions in the SPDE framework. We prove that averaging specific GRFs sampled via the SPDE approach yields genuinely isotropic realisations without domain extension, with the error bounds established in the first part remaining valid.

翻译：高斯过程（GPs）与高斯随机场（GRFs）对于建模空间变化的随机现象至关重要。然而，在高分辨率网格上高效生成相应的实现仍然具有挑战性，尤其是在需要大量实现的情况下。本文提出了两项新颖的贡献。首先，我们提出了一种基于 Dirichlet-Neumann 平均（DNA）的新方法，用于在规则间隔的网格上生成具有各向同性协方差的高斯过程和高斯随机场。DNA 采样方法中离散余弦变换与正弦变换的结合，使得无需修改或填充目标协方差函数即可实现快速评估。虽然这会在协方差中引入误差，但我们的数值实验表明，对于大多数相关应用而言，该误差可忽略不计，这代表了效率与精度之间的权衡。我们为 Matérn 协方差提供了明确的误差估计。第二项贡献将我们的新方法与用于采样 GRF 的随机偏微分方程（SPDE）方法联系起来。我们证明，我们方法中发展的概念同样可以指导 SPDE 框架中边界条件的选择。我们证明了，对通过 SPDE 方法采样的特定 GRF 进行平均，无需扩展域即可产生真正各向同性的实现，且第一部分建立的误差界仍然有效。