Analysis of boundary effects on PDE-based sampling of Whittle-Mat\'ern random fields

TL;DR

This paper analyzes how boundary effects impact PDE-based sampling of Whittle-Matérn Gaussian fields, providing theoretical error bounds and numerical validation for different boundary conditions.

Contribution

It offers a rigorous analysis of covariance errors caused by boundary effects and demonstrates exponential decay of these errors with increasing window size.

Findings

Error decays exponentially with window size

Boundary condition type does not affect error decay rate

Numerical experiments confirm theoretical predictions

Abstract

We consider the generation of samples of a mean-zero Gaussian random field with Mat\'ern covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole $\mathbb{R}^d$, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Mat\'ern field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.