Efficient spatial modelling using the SPDE approach with bivariate splines

TL;DR

This paper introduces a novel approach for spatial modeling using bivariate splines within the SPDE framework, enhancing computational efficiency and flexibility in Gaussian field approximations.

Contribution

It proposes a new class of GF representations with bivariate splines, improving convergence and ease of implementation over traditional finite element methods.

Findings

Faster convergence with higher order bivariate splines

Enhanced computational efficiency demonstrated in simulations

Flexible extension to non-stationary fields

Abstract

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to non-stationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency.