A note on the relationship between PDE-based precision operators and Matérn covariances

TL;DR

The note clarifies how Matérn covariance parameters map to PDE-based precision operators within the hIPPYlib framework. It shows that the Matérn field can be represented as the solution of a fractional SPDE $(\kappa^2 - \Delta)^{\alpha/2} u = s W$ with $\alpha = \nu + d/2$, and provides explicit mappings for the biLaplacian priors in 2D and 3D through the coefficients $\gamma$ and $\delta$, including special cases and domain-boundary considerations. It further discusses extensions to spatially varying coefficients and anisotropy, and presents numerical experiments in 1D and 2D that illustrate boundary effects, the impact of Robin vs Neumann conditions, and the orientation of anisotropic correlation structures. The results guide practical implementation of Matérn-type priors on bounded domains and with direction-dependent correlation in the hIPPYlib framework, linking theoretical parameters to finite-element discretizations and boundary treatments.

Abstract

The purpose of this technical note is to summarize the relationship between the marginal variance and correlation length of a Gaussian random field with Matérn covariance and the coefficients of the corresponding partial-differential-equation (PDE)-based precision operator.

Figures (6)

• Figure 1: Marginal variance (left) and correlation length structure (right) for the biLaplacian prior with Robin boundary conditions in 1D. On the left one can see the effects of the boundary on the marginal variance; in the interior of the domain the marginal variance is close to $4$, but near the boundary it decreases to $\sim2.8$. On the right, the correlation is shown at the point $x =0.5$; the effects of the boundary are limited as the correlation decreases by a factor of $10$ a distance of $\rho =0.25$ away.
• Figure 2: Marginal variance (left) and correlation length structure (right) for the biLaplacian prior with homogeneous Neumann boundary conditions in 1D. On the left one can see the effects of the boundary on the marginal variance; in the interior of the domain the marginal variance is close to $4$, but near the boundary it increases to $\sim8$. Additionally the marginal variance in the interior is more substantially polluted in the interior than in Figure \ref{['fig:1d_robin_var_corr']}. On the right, the correlation is shown at the point $x =0.75=1.0 - \rho$. In this instance the boundary effects substantially effect the correlation structure near to the boundary.
• Figure 3: Marginal variance (left) and correlation length structure (right) for the biLaplacian prior with Robin boundary conditions in 2D. On the left one can see the effects of the boundary on the marginal variance; in the interior of the domain the marginal variance is close to $4$, but near the boundary it is reduced a small amount. On the right, the correlation is shown at the point $\mathbf{x} = (0.75,0.75) = (1-\rho,1-\rho)$; the effects of the boundary are limited as the correlation decreases by a factor of $10$ a distance of $\rho =0.25$ away.
• Figure 4: Marginal variance (left) and correlation length structure (right) for the biLaplacian prior with homogeneous Neumann boundary conditions in 2D. On the left one can see the effects of the boundary on the marginal variance; in the interior of the domain the marginal variance is close to $4$, but near the boundary it increases, reaching $\sim15$ at the corners. Additionally the marginal variance in the interior is more substantially polluted in the interior than in Figure \ref{['fig:2d_robin_var_corr']}. On the right, the correlation is shown at the point $\mathbf{x} = (0.75,0.75) = (1-\rho,1-\rho)$. In this instance the boundary effects substantially effect the correlation structure near to the boundary.
• Figure 5: Two isotropic samples from the biLaplcian prior with Robin boundary conditions on a $64\times 64$ mesh.