We present a detailed mathematical description of the connection between Gaussian processes with covariance operators defined by the Matern covariance function and Gaussian processes with precision (inverse-covariance) operators defined by the Green's functions of a class of elliptic stochastic partial differential equations (SPDEs). We will show that there is an equivalence between these two Gaussian processes when the domain is infinite -- for us, $\mathbb{R}$ or $\mathbb{R}^2$ -- which breaks down when the domain is finite due to the effect of boundary conditions on Green's functions of PDEs. We show how this connection can be re-established using extended domains. We then introduce the semivariogram method for obtaining point estimates of the Matern covariance hyper-parameters, which specifies the Gaussian prior needed for stabilizing the inverse problem. We implement the method on one- and two-dimensional image deblurring test cases to show that it works on practical examples. Finally, we define a Bayesian hierarchical model, assuming hyper-priors on the precision and Matern hyper-parameters, and then sample from the resulting posterior density function using Markov chain Monte Carlo (MCMC), which yields distributional approximations for the hyper-parameters.

中文翻译：

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在贝叶斯逆问题中建模 Whittle-Matérn 先验的半变异函数方法

我们详细描述了具有由 Matern 协方差函数定义的协方差算子的高斯过程与具有由一类椭圆随机偏微分方程 (SPDE) 的格林函数定义的精度（逆协方差）算子的高斯过程之间的联系的详细数学描述。我们将证明，当域是无限的时，这两个高斯过程之间存在等价性——对我们来说，$\mathbb{R}$ 或 $\mathbb{R}^2$——当域是有限的时就会失效由于边界条件对偏微分方程格林函数的影响。我们展示了如何使用扩展域重新建立这种连接。然后我们介绍了用于获得 Matern 协方差超参数的点估计的半变异函数方法，它指定了稳定逆问题所需的高斯先验。我们在一维和二维图像去模糊测试用例上实现了该方法，以表明它适用于实际示例。最后，我们定义了一个贝叶斯分层模型，假设精度和 Matern 超参数的超先验，然后使用马尔可夫链蒙特卡罗 (MCMC) 从得到的后验密度函数中采样，这会产生超参数的分布近似值。