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

TL;DR

This paper introduces a novel mesh-free Boltzmann-Gibbs random field model using smoothed particle hydrodynamics to define local interactions, enabling seamless extension from discrete data to continuum fields.

Contribution

It develops a new Boltzmann-Gibbs model with local interactions based on smoothed kernel approximations, providing explicit mesh-free precision functions for continuum field modeling.

Findings

Derived explicit mesh-free precision functions for Gaussian kernels.

Connected the model with Gaussian Markov random fields and Matern fields.

Extended the model from discrete data to continuum fields.

Abstract

Boltzmann-Gibbs random fields are defined in terms of the exponential expression exp(-H), where H is a suitably defined energy functional of the field states x(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 Mat\'{e}rn field with $\nu=1$ are established.