Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

Boltzmann–Gibbs random fields are defined in terms of the exponential expression $\exp \left (-\mathcal {H}\right )$, where $\mathcal {H}$ is a suitably defined energy functional of the field states $x(\mathbf {s})$. This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with $\nu =1$ are established.