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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\u2013Richardson 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.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae098","type":"journal-article","created":{"date-parts":[[2024,9,18]],"date-time":"2024-09-18T19:23:37Z","timestamp":1726687417000},"page":"457-479","source":"Crossref","is-referenced-by-count":6,"title":["Probabilistic Richardson extrapolation"],"prefix":"10.1093","volume":"87","author":[{"given":"Chris J","family":"Oates","sequence":"first","affiliation":[{"name":"School of Mathematics, Statistics and Physics, Newcastle University , Newcastle upon Tyne ,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Toni","family":"Karvonen","sequence":"additional","affiliation":[{"name":"School of Engineering Sciences, Lappeenranta\u2013Lahti University of Technology LUT , Lappeenranta ,","place":["Finland"]},{"name":"Department of Mathematics and Statistics, University of Helsinki , Helsinki ,","place":["Finland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aretha L","family":"Teckentrup","sequence":"additional","affiliation":[{"name":"School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh , Edinburgh ,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marina","family":"Strocchi","sequence":"additional","affiliation":[{"name":"National Heart and Lung Institute, Imperial College London , London ,","place":["UK"]},{"name":"School of Biomedical Engineering and Imaging Sciences, King\u2019s College London , London ,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven A","family":"Niederer","sequence":"additional","affiliation":[{"name":"National Heart and Lung Institute, Imperial College London , London ,","place":["UK"]},{"name":"School of Biomedical Engineering and Imaging Sciences, King\u2019s College London , London ,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,12,26]]},"reference":[{"key":"2025041100301311700_qkae098-B1","doi-asserted-by":"publisher","first-page":"133115","DOI":"10.33737\/jgpps\/133115","article-title":"Towards the large-eddy simulation of a full engine: Integration of a 360 azimuthal degrees fan, compressor and combustion chamber. 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