# Probabilistic Richardson Extrapolation

arXiv: https://arxiv.org/abs/2401.07562 · HTML: https://mlsanity.com/paper/2401.07562/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2024.

## Built on by (most cited first, 1 of 1)

- [Extrapolating from Regularised Solutions for Solving Ill-Conditioned Linear Systems in Machine Learning](https://mlsanity.com/paper/2606.30328/) (2026)

## Abstract (arXiv)

For over a century, extrapolation methods have provided a powerful tool to improve the convergence order of a numerical method. However, these tools are not well-suited to modern computer codes, where multiple continua are discretised and convergence orders are not easily analysed. To address this challenge we present a probabilistic perspective on Richardson extrapolation, a point of view that unifies classical extrapolation methods with modern multi-fidelity modelling, and handles uncertain convergence orders by allowing these to be statistically estimated. The approach is developed using Gaussian processes, leading to Gauss-Richardson Extrapolation (GRE). Conditions are established under which extrapolation using the conditional mean achieves a polynomial (or even an exponential) speed-up compared to the original numerical method. Further, the probabilistic formulation unlocks the possibility of experimental design, casting the selection of fidelities as a continuous optimisation problem which can then be (approximately) solved. A case-study involving a computational cardiac model demonstrates that practical gains in accuracy can be achieved using the GRE method.