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Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation

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Abstract

This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the one-dimensional stochastic Allen–Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to \(\frac{1}{2}\), like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.

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References

  1. Allen, S., Cahn, J.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 27(6), 1085–1095 (1979)

    Article  Google Scholar 

  2. Andersson, A., Kruse, R., Larsson, S.: Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 4(1), 113–149 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Andersson, A., Larsson, S.: Weak convergence for a spatial approximation of the nonlinear stochastic heat equation. Math. Comput. 85(299), 1335–1358 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Becker, S., Gess, B., Jentzen, A., Kloeden, P.E.: Strong convergence rates for explicit space time discrete numerical approximations of stochastic Allen–Cahn equations (2017). arXiv preprint arXiv:1711.02423

  5. Bréhier, C.-E.: Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs. J. Complex. 56 (2019)

  6. Bréhier, C.-E., Cui, J., Hong, J.: Strong convergence rates of semi-discrete splitting approximations for stochastic Allen-Cahn equation. IMA J. Numer. Anal. 39, 2096–2134 (2018). https://doi.org/10.1093/imanum/dry052

    Article  MATH  Google Scholar 

  7. Bréhier, C.-E., Debussche, A.: Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient. J. Math. Pures Appl. 119, 193–254 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bréhier, C.-E., Goudenège, L.: Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete Contin. Dyn. Syst. B 24(8), 4169–4190 (2019). https://doi.org/10.3934/dcdsb.2019077

    Article  MathSciNet  MATH  Google Scholar 

  9. Cerrai, S.: Second Order PDE’s in Finite and Infinite Dimension. Lecture Notes in Mathematics. A probabilistic approach, vol. 1762. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  10. Conus, D., Jentzen, A., Kurniawan, R.: Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Ann. Appl. Probab. 29(2), 653–716 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Da Prato, G., Debussche, A.: An integral inequality for the invariant measure of a stochastic reaction–diffusion equation. J. Evol. Equ. 17(1), 197–214 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  13. Debussche, A.: Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comput. 80(273), 89–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Debussche, A., Printems, J.: Weak order for the discretization of the stochastic heat equation. Math. Comput. 78(266), 845–863 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grisvard, P.: Caractérisation de quelques espaces d’interpolation. Arch. Ration. Mech. Anal. 25(1), 40–63 (1967)

    Article  MATH  Google Scholar 

  16. Hefter, M., Jentzen, A., Kurniawan, R.: Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces (2016). arXiv preprint arXiv:1612.03209

  17. Jentzen, A., Kloeden, P.E.: Taylor Approximations for Stochastic Partial Differential Equations. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 83. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011)

    Book  MATH  Google Scholar 

  18. Jentzen, A., Kurniawan, R.: Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients (2015). arXiv:1501.03539

  19. Kopec, M.: Quelques contributions à l’analyse numérique d’équations stochastiques. PhD thesis, Ecole normale supérieure de Rennes-ENS Rennes (2014)

  20. Kovács, M., Larsson, S., Lindgren, F.: On the backward Euler approximation of the stochastic Allen–Cahn equation. J. Appl. Probab. 52(2), 323–338 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kovács, M., Larsson, S., Lindgren, F.: On the discretisation in time of the stochastic Allen-Cahn equation. Math. Nachrichten 291(5–6), 966–995 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kruse, R.: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Lecture Notes in Mathematics, vol. 2093. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  23. Liu, Z., Qiao, Z.: Wong-Zakai approximations of stochastic Allen-Cahn equation. Int. J. Numer. Anal. Mod. 16, 681–694 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Liu, Z., Qiao, Z.: Strong approximation of monotone stochastic partial differential equations driven by white noise. IMA J. Numer. Anal. (2019). https://doi.org/10.1093/imanum/dry088

  25. Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York (2014)

    Book  MATH  Google Scholar 

  26. Majee, A., Prohl, A.: Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math. 18(2), 297–311 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications (New York), 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  28. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Book  MATH  Google Scholar 

  29. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)

    MATH  Google Scholar 

  30. Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10(4), 545–551 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, X.: Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete Contin. Dyn. Syst. 36(1), 481–497 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, X.: An efficient explicit full discrete scheme for strong approximation of stochastic Allen–Cahn equation (2018). arXiv preprint arXiv:1802.09413

  33. Wang, X., Gan, S.: Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise. J. Math. Anal. Appl. 398(1), 151–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous referees for their remarks and suggestions. They also want to thank Arnaud Debussche for discussions, and for suggesting the approach to prove Lemma 4.4. They also wish to thank Jialin Hong and Jianbao Cui for helpful comments and suggestions to improve the presentation of the manuscript.

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Authors and Affiliations

  1. CNRS, UMR5208, Institut Camille Jordan, Univ Lyon, Université Claude Bernard Lyon 1, 69622, Villeurbanne, France

    Charles-Edouard Bréhier

  2. CNRS - FR3487, Fédération de Mathématiques de CentraleSupélec, CentraleSupélec, Université Paris-Saclay, 3 Rue Joliot Curie, 91190, Gif-sur-Yvette, France

    Ludovic Goudenège

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  1. Charles-Edouard Bréhier
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  2. Ludovic Goudenège
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Correspondence to Charles-Edouard Bréhier.

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Bréhier, CE., Goudenège, L. Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation. Bit Numer Math 60, 543–582 (2020). https://doi.org/10.1007/s10543-019-00788-x

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