Probabilistic Richardson Extrapolation
Chris. J. Oates, Toni Karvonen, Aretha L. Teckentrup, Marina Strocchi, Steven A. Niederer
TL;DR
This paper introduces Gauss-Richardson Extrapolation (GRE), a probabilistic framework that unifies classical Richardson extrapolation with modern multi-fidelity modelling by modeling the discretisation map $f(\mathbf{x})$ with a numerical-analysis-informed Gaussian process. By conditioning on a design of fidelities and using the conditional mean as the extrapolator, GRE achieves polynomial or exponential speed-ups to estimate the continuum limit $f(\mathbf{0})$, while providing credible intervals and thus principled uncertainty quantification. It establishes higher-order convergence guarantees under finite and infinite smoothness, develops objective and Bayesian-style experimental design for fidelity selection, and extends to unknown convergence orders, multidimensional outputs, and additional degrees of freedom. The methodology is validated on a computational cardiac model, showing practical accuracy gains under realistic budgets and offering a generalisable framework for accelerating costly simulations with quantified uncertainty. The work highlights the potential of probabilistic numerics to guide extrapolation and design decisions in high-cost simulations across scientific domains.
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 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.