Lower Bounds for Weak Approximation Errors for Spatial Spectral Galerkin Approximations of Stochastic Wave Equations
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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.
KeywordsStochastic wave equations Weak convergence Lower bounds Essentially sharp convergence rates Spectral Galerkin approximations
Mathematics Subject Classification classes60H15 65C30
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”.
- 1.Conus, D., Jentzen, A., Kurniawan, R.: Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. ArXiv e-prints (Aug 2014), 59 pages. arXiv: 1408.1108 [math.PR]. Accepted in Ann. Appl. Probab.
- 2.Davie, A.M., Gaines, J.G.: Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comput. 70, 233, 121–134 (2001). ISSN 0025-5718. https://doi.org/10.1090/S0025-5718-00-01224-2
- 3.Hausenblas, E.: Weak approximation of the stochastic wave equation. J. Comput. Appl. Math. 235, 1, 33–58 (2010). ISSN 0377-0427. https://doi.org/10.1016/j.cam.2010.03.026
- 4.Jacobe de Naurois, L., Jentzen, A., Welti, T.: Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. ArXiv e-prints (Aug 2015), 27 pages. arXiv: 1508.05168 [math.PR]. Accepted in Appl. Math. Optim.
- 5.Jentzen, A., Kurniawan, R.: Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. ArXiv e- prints (Jan 2015), 51 pages. arXiv: 1501.03539 [math.PR]
- 6.Kovács, M., Larsson, S., Lindgren, F.: Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise. BIT 52, 1, 85–108 (2012). ISSN 0006-3835. https://doi.org/10.1007/s10543-011-0344-2
- 7.Kovács, M., Larsson, S., Lindgren, F.: Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes. BIT 53, 2, 497–525 (2013). ISSN 0006-3835Google Scholar
- 8.Kovács, M., Lindner, F., Schilling, R.L.: Weak convergence of finite element approximations of linear stochastic evolution equations with additive Lévy noise. SIAM/ASA J. Uncertain. Quantif. 3, 1, 1159–1199 (2015). ISSN 2166-2525. https://doi.org/10.1137/15M1009792
- 9.Müller-Gronbach, T., Ritter, K.: Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7, 2, 135–181 (2007). ISSN 1615-3375. https://doi.org/10.1007/s10208-005-0166-6
- 10.Müller-Gronbach, T., Ritter, K., Wagner, T.: Optimal pointwise approximation of infinite-dimensional Ornstein-Uhlenbeck processes. Stoch. Dyn. 8, 3, 519–541 (2008). ISSN 0219-4937. https://doi.org/10.1142/S0219493708002433
- 11.Sell, G.R., You, Y.: Dynamics of evolutionary equations, vol. 143. Applied Mathematical Sciences. Springer, New York, 2002, xiv+670. ISBN 0-387-98347-3. https://doi.org/10.1007/978-1-4757-5037-9
- 12.Wang, X.: An exponential integrator scheme for time discretization of nonlinear stochastic wave equation. J. Sci. Comput. 64, 1, 234–263 (2015). ISSN 0885-7474. https://doi.org/10.1007/s10915-014-9931-0