Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

Bréhier, Charles-Edouard

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 151-175

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

Classification : 60H35, 65C30, 60H15
Keywords: Stochastic partial differential equations, exponential integrators, tamed scheme, invariant distribution

[1] M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner and D. Salimova, Strong and weak divergence of exponential and linear-implicit euler approximations for SPDEs with superlinearly growing nonlinearities. Preprint (2019). | arXiv

[2] S. Boyaval, S. Martal and J. Reygner, Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law. IMA J. Numer. Anal. (2021). DOI: | DOI | MR

[3] C.-E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise. Potential Anal. 40 (2014) 1–40. | MR | Zbl | DOI

[4] C.-E. Bréhier, Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs. J. Complexity 56 (2020) 101424. | MR | DOI

[5] C.-E. Bréhier and L. Goudenège, Weak convergence rates of splitting schemes for the stochastic Allen-Cahn equation. BIT 60 (2020) 543–582. | MR | DOI

[6] C.-E. Bréhier and M. Kopec, Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme. IMA J. Numer. Anal. 37 (2017) 1375–1410. | MR

[7] C.-E. Bréhier and G. Vilmart, High order integrator for sampling the invariant distribution of a class of parabolic stochastic PDEs with additive space-time noise. SIAM J. Sci. Comput. 38 (2016) A2283–A2306. | MR | DOI

[8] M. Cai, S. Gan and X. Wang, Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise. J. Sci. Comput. 86 (2021) 1–30. | MR

[9] S. Cerrai, Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Vol. 1762 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2001). | MR | Zbl | DOI

[10] Z. Chen, S. Gan and X. Wang, A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equations. Appl. Numer. Math. 157 (2020) 135–158. | MR | DOI

[11] J. Cui and J. Hong, Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient. SIAM J. Numer. Anal. 57 (2019) 1815–1841. | MR | DOI

[12] J. Cui, J. Hong and L. Sun, Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients. Stoch. Process. Appl. 134 (2021) 55–93. | MR | DOI

[13] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems. Vol 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1996). | MR | Zbl | DOI

[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Vol. 152 of Encyclopedia of Mathematics and its Applications, 2nd edition. Cambridge University Press, Cambridge (2014). | MR | Zbl | DOI

[15] I. Gyöngy, S. Sabanis and D. Šiška, Convergence of tamed Euler schemes for a class of stochastic evolution equations. Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016) 225–245. | MR

[16] J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations: Numerical Approximations and Symplectic Structures. Vol. 2251 of Lecture Notes in Mathematics. Springer, Singapore (2019). | MR | DOI

[17] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 236 (2015) v+99. | MR

[18] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011) 1563–1576. | MR | Zbl

[19] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (4) 1611–1641. | MR | Zbl

[20] A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649–667. | MR | Zbl

[21] G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York, 2014. | MR | Zbl | DOI

[22] G. N. Milstein and M. V. Tretyakov, Computing ergodic limits for Langevin equations. Phys. D 229 (2007) 81–95. | MR | Zbl | DOI

[23] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. 2nd edition. Johann Ambrosius Barth, Heidelberg (1995). | MR | Zbl

[24] X. Wang, An efficient explicit full discrete scheme for strong approximation of stochastic Allen-Cahn equation. Stoch. Processes App. 130 (2020) 6271–6299. | MR | DOI