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Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

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Abstract

This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen–Cahn equations with space-time white noise which have been obtained in a recent previous work of the authors of this article.

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Acknowledgements

This work has been partially supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”. The third author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Muenster: Dynamics-Geometry-Structure. Financial support by the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is acknowledged.

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Authors and Affiliations

  1. RiskLab, Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Sebastian Becker

  2. Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Benjamin Gess

  3. Faculty of Mathematics, University of Bielefeld, 33615, Bielefeld, Germany

    Benjamin Gess

  4. SAM, Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Arnulf Jentzen

  5. Faculty of Mathematics and Computer Science, University of Münster, 48149, Münster, Germany

    Arnulf Jentzen

  6. Mathematics Institute, Goethe University Frankfurt, 60325, Frankfurt am Main, Germany

    Peter E. Kloeden

Authors
  1. Sebastian Becker
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  2. Benjamin Gess
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  3. Arnulf Jentzen
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  4. Peter E. Kloeden
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Correspondence to Sebastian Becker.

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Communicated by David Cohen.

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Becker, S., Gess, B., Jentzen, A. et al. Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. Bit Numer Math 60, 1057–1073 (2020). https://doi.org/10.1007/s10543-020-00807-2

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  • DOI: https://doi.org/10.1007/s10543-020-00807-2

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