对基于PDE的Whittle-Matérn随机场采样的边界效应分析
Analysis of boundary effects on PDE-based sampling of Whittle-Matérn random fields
AI总结：
本文研究了在有限域上求解具有Matérn协方差函数的零均值高斯随机场样本时，边界效应的影响，通过窗口技术引入合适的边界条件，并分析不同边界条件下的误差衰减特性。
摘要：
我们考虑了生成具有Matérn协方差函数的零均值高斯随机场样本的问题。每个样本都需要求解带有高斯白噪声驱动项的微分方程，该方程定义在有限计算域上。这引入了不想要的边界效应，因为原来的随机偏微分方程是在整个$\mathbb{R}^d$上定义的，没有边界条件。我们采用窗口技术，将计算域嵌入到更大的域中，并在扩展域上假设方便的边界条件。为了减轻人工边界带来的污染，在数值研究中建议选择窗口大小至少等于Matérn场的相关长度。我们为窗口技术引入的协方差误差进行了严格分析，考虑了齐次狄利克雷、齐次奈曼和周期边界条件。我们证明误差随着窗口大小指数衰减，与边界条件类型无关。我们在1D和2D空间中进行了数值实验，验证了我们的理论结果。
英文摘要：
We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole $\mathbb{R}^d$, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.
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