Dirichlet-Neumann Averaging: The DNA of Efficient Gaussian Process Simulation
Journal:
arXiv
Published Date:
Dec 10, 2024
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 Mat\'ern 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.
