Abstract
We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.
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Acknowledgements
I would like to thank Professor Paweł Przybyłowicz for many inspiring discussions while preparing this manuscript. I am also very grateful to two anonymous reviewers for their valuable comments and suggestions that allowed me to improve the quality of the paper.
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This research was funded in whole or in part by the National Science Centre, Poland, under project 2021/41/N/ST1/00135. For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
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Communicated by: Anthony Nouy
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Bochacik, T. On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes. Adv Comput Math 49, 14 (2023). https://doi.org/10.1007/s10444-023-10012-8
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DOI: https://doi.org/10.1007/s10444-023-10012-8
Keywords
- Exceptional set
- Confidence region
- Noisy information
- Randomized algorithm
- Explicit and implicit Euler schemes
- Two-stage Runge-Kutta scheme