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Probabilistic Richardson extrapolation.

Chris J Oates1, Toni Karvonen2,3, Aretha L Teckentrup4

  • 1School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, UK.

Journal of the Royal Statistical Society. Series B, Statistical Methodology
|April 14, 2025
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Summary
This summary is machine-generated.

Gauss-Richardson Extrapolation (GRE) offers a probabilistic approach to improve numerical methods. This method handles complex computer codes and uncertain convergence orders, achieving significant speed-ups and accuracy gains.

Keywords:
Bayesian statisticsGaussian processmulti-fidelity modellingreproducing kerneluncertainty quantification

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Area of Science:

  • Numerical Analysis
  • Computational Science
  • Applied Mathematics

Background:

  • Extrapolation methods enhance numerical method convergence orders.
  • Traditional methods struggle with modern complex computer codes and uncertain convergence.
  • Multi-fidelity modeling presents challenges in analyzing convergence orders.

Purpose of the Study:

  • Introduce a probabilistic perspective on Richardson extrapolation.
  • Unify classical extrapolation with multi-fidelity modeling.
  • Develop a method to handle uncertain convergence orders statistically.

Main Methods:

  • Developed Gauss-Richardson Extrapolation (GRE) using Gaussian processes.
  • Established conditions for polynomial or exponential speed-ups.
  • Formulated experimental design as a continuous optimization problem.

Main Results:

  • Gauss-Richardson Extrapolation (GRE) provides a probabilistic framework.
  • GRE statistically estimates uncertain convergence orders.
  • Demonstrated practical accuracy gains in a computational cardiac model.

Conclusions:

  • GRE unifies classical extrapolation with multi-fidelity modeling.
  • The probabilistic approach enables statistical estimation of convergence orders.
  • GRE offers a powerful tool for enhancing numerical methods in complex simulations.