# The rational SPDE approach for Gaussian random fields with general smoothness

arXiv: https://arxiv.org/abs/1711.04333 · HTML: https://mlsanity.com/paper/1711.04333/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2017.

## Built on by (most cited first, 7 of 7)

- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Gaussian Process Regression under Computational and Epistemic Misspecification](https://mlsanity.com/paper/2312.09225/) (2023)
- [Efficient methods for Gaussian Markov random fields under sparse linear constraints](https://mlsanity.com/paper/2106.01712/) (2021)
- [Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis](https://mlsanity.com/paper/2305.13879/) (2023)
- [Modeling High-Resolution Spatio-Temporal Wind with Deep Echo State Networks and Stochastic Partial Differential Equations](https://mlsanity.com/paper/2412.07265/) (2024)
- [Sampling as Bandits: Evaluation-Efficient Design for Black-Box Densities](https://mlsanity.com/paper/2509.01437/) (2025)
- [Bayesian Optimization on Networks](https://mlsanity.com/paper/2510.27643/) (2025)

## Abstract (arXiv)

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form Lβu = W, where W is Gaussian white noise, L is a second-order differential operator, and β>0 is a parameter that determines the smoothness of u. However, this approach has been limited to the case 2β∈mathbbN, which excludes several important models and makes it necessary to keep β fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension d∈mathbbN is applicable for any β>d/4, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function x-β to approximate u. For the resulting approximation, an explicit rate of convergence to u in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case 2β∈mathbbN. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including β.