Dirichlet-Neumann Averaging: The DNA of Efficient Gaussian Process Simulation

TL;DR

This work addresses efficient generation of Gaussian random field realisations with isotropic covariances on high-resolution grids. It introduces Dirichlet-Neumann Averaging (DNA), a periodisation-based sampling framework using Discrete Sine and Cosine Transforms to produce genuinely isotropic covariances via a periodised function with controlled error, and provides explicit Matérn covariance error bounds that decay exponentially with the scaling parameter $$\alpha$$. The authors further connect DNA to the SPDE sampling paradigm, showing that averaging SPDE solutions with different boundary conditions yields isotropic fields without domain extension, while preserving the same error guarantees. Numerical experiments demonstrate that DNA achieves negligible covariance error in practice, offers favorable parallelisation and memory characteristics, and can outperform traditional padding-based circulant embedding and oversampling in SPDE methods, making it a scalable and robust approach for GP/GRF simulation on large grids.

Abstract

Gaussian processes (GPs) and Gaussian random fields (GRFs) are essential for modelling spatially varying stochastic phenomena. Yet, the efficient generation of corresponding realisations on high-resolution grids remains challenging, particularly when a large number of realisations are required. This paper presents two novel contributions. First, we propose a new methodology based on Dirichlet-Neumann averaging (DNA) to generate GPs and GRFs with isotropic covariance on regularly spaced grids. The combination of discrete cosine and sine transforms in the DNA sampling approach allows for rapid evaluations without the need for modification or padding of the desired covariance function. While this introduces an error in the covariance, our numerical experiments show that this error is negligible for most relevant applications, representing a trade-off between efficiency and precision. We provide explicit error estimates for Matérn covariances. The second contribution links our new methodology to the stochastic partial differential equation (SPDE) approach for sampling GRFs. We demonstrate that the concepts developed in our methodology can also guide the selection of boundary conditions in the SPDE framework. We prove that averaging specific GRFs sampled via the SPDE approach yields genuinely isotropic realisations without domain extension, with the error bounds established in the first part remaining valid.

Figures (7)

• Figure 1: Three periodisations of the Ma-térn covariance function $\varphi$ with parameters $\nu = 2$ and $\ell = 2/5$, corresponding to the sampling algorithms discussed in Section \ref{['sec:sampFr']}. The scaling parameter is $\alpha = 1$.
• Figure 2: Overhead for Ma-térn covariance with smoothness $\nu$ on $D=(0,1)$ with $n=1500$ grid points. Minimal embedding determined via bisection; missing values indicate padding exceeding 256.
• Figure 3: Heatmaps of target (top left) and empirical covariances for Ma-térn fields ($\nu = 2$, $\ell = 0.15$) with $N = 5 \cdot 10^5$ realisations. Clockwise from top right: homogeneous Neumann, homogeneous Dirichlet, periodic boundary conditions.
• Figure 4: Empirical marginal variances for a GRF with Ma-térn covariance function with $\nu = 2.0$ and $\ell = 0.15$, estimated using $N = 5 \cdot 10^5$ realisations. Values correspond to diagonals of the empirical covariance matrices in Figure \ref{['fig:covBndryConditions']}.
• Figure 5: Empirical marginal variances for the $2^d = 4$ combinations of boundary conditions used for DNA sampling, along with the resulting empirical marginal variance of the DNA GRF. Parameters for the Ma-térn covariance: $\nu = 1.5$, $\ell = 0.2$; estimated from $N = 10^4$ realisations on a $150 \times 150$ grid.

Theorems & Definitions (14)

• Definition 1.1: Isotropic Covariance Function
• Definition 1.3: Periodisation
• Definition 1.4: Periodised Random Field
• Definition 2.1: Dirichlet-Neumann Averaged Gaussian Random Field
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Remark 2.6
• Proposition 2.7