Abstract
This paper presents a study of the approximation error corresponding to a symplectic scheme of weak order one for a stochastic autonomous Hamiltonian system. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. An expansion of the weak error and expansions of the ergodic averages and of the invariant measures associated with the numerical scheme are obtained in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation.
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Burrage, K., Burrage, P.: Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise. J. Comput. Appl. Math. 236, 3920–3930 (2012)
Milstein, G.N., Repin, Y.M., Tretyakov, M.V.: Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. 39(6), 2066–2088 (2002)
Milstein, G.N., Repin, Y.M., Tretyakov, M.V.: Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal. 40(4), 1583–1604 (2002)
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2004)
Hong, J., Sun, L., Wang, X.: High order conformal symplectic and ergodic schemes for the stochastic Langevin equation via generating functions. SIAM J. Numer. Anal. 55(6), 3006–3029 (2017)
Sun, L., Wang, L.: Stochastic symplectic methods based on the Pade approximations for linear stochastic Hamiltonian systems. J. Comput. Appl. Math. 311, 439–456 (2017)
Anton, C., Deng, J., Wong, Y.: Weak symplectic schemes for stochastic Hamiltonian equations. Electron. Trans. Numer. Anal. 43, 1–20 (2014)
Deng, J., Anton, C., Wong, Y.: High-order symplectic schemes for stochastic Hamiltonian systems. Commun. Comput. Phys. 16(1), 169–200 (2014)
Sanz-Serna, J.M., Calvo, M.: Numerical Hamiltonian problems. Chapman and Hall, London (1994)
Reich, S.: Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36, 1549–1570 (1999)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)
Debussche, A., Faou, E.: Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50(3), 1735–1752 (2012)
Shardlow, T.: Modified equations for stochastic differential equations. BIT Numer. Math. 46(1), 111–125 (2006)
Abdulle, A., Cohen, D., Vilmart, G., Zygalakis, K.: High weak order methods for stochastic differential equations based on modified equations. SIAM J. Sci. Comput. 34(3), 1800–1823 (2012)
Zygalakis, K.: On the existence and applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput. 33(1), 102–130 (2015)
Abdulle, A., Vilmart, G., Zygalakis, K.: Long time accuracy of Lie–Trotter splitting methods for Langevin dynamics. SIAM J. Sci. Comput. 53(1), 1–16 (2015)
Kopec, M.: Weak backward error analysis for Langevin process. BIT Numer. Math. 55(4), 1057–1103 (2015)
Kopec, M.: Weak backward error analysis for overdamped Langevin processes. IMA J. Numer. Anal. 35(2), 583–614 (2015)
Wang, L., Hong, J., Sun, L.: Modified equations for weakly convergent stochastic symplectic schemes via their generating functions. BIT Numer. Math. 56, 1131–1162 (2016)
Milstein, G.N., Tretyakov, M.V.: Quasi-symplectic methods for Langevin-type equations. IMA J. Numer. Anal. 23, 593–626 (2003)
Talay, D.: Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stoch. Stoch. Rep. 29(1), 13–36 (1990)
Talay, D.: Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Proc. Relat. Fields 8, 1–36 (2002)
Mattingly, J., Stuart, A., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 2(101), 185–232 (2002)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 483–509 (1990)
Anton, C., Wong, Y., Deng, J.: On global error of symplectic schemes for stochastic Hamiltonian systems. Int. J. Numer. Anal. Model. Ser. B 4(1), 80–93 (2013)
Mattingly, J., Stuart, A., Tretyakov, M.: Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 28(2), 552–577 (2010)
Pardoux, E., Veretennikov, Y.: On the Poisson equation and diffusion approximation. Ann. Probab. 29(3), 1061–1085 (2001)
Meyn, S.P., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (1993)
Hutzenthaler, M., Jentzen, A., Kloeden, P.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with nonglobally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 467, 1563–1576 (2011)
Hutzenthaler, M., Jentzen, A., Kloeden, P.: Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)
Tretyakov, M.V., Zhang, Z.: A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. 51(6), 3135–3162 (2013)
Mao, X., Szpruch, L.: Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients. J. Comput. App. Math. 238, 14–28 (2013)
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: Hennequin, P. (ed.) École d’Été de Probabilités de Saint-Flour XII-1982. Lecture Notes in Mathematics, pp. 143–303. Springer, Berlin (1984)
Hasminskii, R.Z.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2012)
Zhu, C., Yin, G.: Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46(4), 1155–1179 (2007)
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Communicated by David Cohen.
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This research was supported by the NSERC grant DDG-2015-00041.
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Anton, C. Weak backward error analysis for stochastic Hamiltonian Systems. Bit Numer Math 59, 613–646 (2019). https://doi.org/10.1007/s10543-019-00747-6
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DOI: https://doi.org/10.1007/s10543-019-00747-6