Semivariogram methods for modeling Whittle–Matérn priors in Bayesian inverse problems

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.