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Lower Bounds for Weak Approximation Errors for Spatial Spectral Galerkin Approximations of Stochastic Wave Equations

  • Ladislas Jacobe de Naurois
  • Arnulf Jentzen
  • Timo Welti
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical approximation methods are needed in order for mathematical modelling with stochastic wave equations to become relevant for real world applications. Therefore, the numerical analysis of convergence rates for such numerical approximation processes is required. A recent article by the authors proves upper bounds for weak errors for spatial spectral Galerkin approximations of a class of semilinear stochastic wave equations. The findings there are complemented by the main result of this work, that provides lower bounds for weak errors which show that in the general framework considered the established upper bounds can essentially not be improved.

Keywords

Stochastic wave equations Weak convergence Lower bounds Essentially sharp convergence rates Spectral Galerkin approximations 

Mathematics Subject Classification classes

60H15 65C30 

Notes

Acknowledgements

This project has been partially supported through the ETH Research Grant ETH-47 15-2 “Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with Lévy noise”.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ETH ZürichZürichSwitzerland

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