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arXiv · 2401.07562

Probabilistic Richardson Extrapolation

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

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Chris. J. Oates, Toni Karvonen, Aretha L. Teckentrup, Marina Strocchi, Steven A. Niederer. 2024-01-15. Probabilistic Richardson Extrapolation. https://arxiv.org/abs/2401.07562

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