The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Abstract

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.

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

“…As mentioned in the above paragraph, the ensuing formulation has a direct connection with GMRFs. These properties (sparse precision matrix and GMRF connection) are shared by the popular stochastic partial differential equation (SPDE) approach [32], recently reviewed in [41]. Note that the BG-LAP2 model defined in Sec-tion 3 can be derived from an SPDE with a suitable differential operator which involves the first and the second power of the half-Laplacian [19].…”

Minimum Information Variability in Linear Langevin Systems via Model Predictive Control

“…In this study, the use of deterministic descriptions of the first two statistical moments over time has facilitated the control, prediction, and simulation of the PDF, owing to the nature of the Langevin equations under consideration. However, in scenarios involving pure data or more complex stochastic differential equations, estimating time-varying PDFs may require inference methods [62] or stochastic simulations [63].…”