Probabilistic Richardson extrapolation

Chris J Oates, Toni Karvonen, Aretha L Teckentrup, Marina Strocchi, Steven A Niederer

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2025 · Volume 87 · Issue 2

Abstract

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 discretized 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 . 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 optimization 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.

Keywords

Bayesian statistics, Gaussian process, multi-fidelity modelling, reproducing kernel, uncertainty quantification

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