Generating functions for stochastic symplectic methods

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March 2014, 34(3): 1211-1228. doi: 10.3934/dcds.2014.34.1211

Generating functions for stochastic symplectic methods

Lijin Wang ^1, and Jialin Hong ^2,

1.	School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
2.	State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190

Received November 2012 Revised April 2013 Published August 2013

Abstract
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Symplectic integration of stochastic Hamiltonian systems is a developing branch of stochastic numerical analysis. In the present paper, a stochastic generating function approach is proposed, based on the derivation of stochastic Hamilton-Jacobi PDEs satisfied by the generating functions, and a method of approximating solutions to them. Thus, a systematic approach of constructing stochastic symplectic methods is provided. As validation, numerical tests on several stochastic Hamiltonian systems are performed, where some symplectic schemes are constructed via stochastic generating functions. Moreover, generating functions for some known stochastic symplectic mappings are given.

Keywords: Hamilton-Jacobi PDEs., symplecticity, generating functions, Stochastic Hamiltonian systems, stochastic symplectic methods.

Mathematics Subject Classification: 65P10, 65C30, 65C20, 65H1.

Citation: Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211

References:

[1]	K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38 (2000), 1626-1646. doi: 10.1137/S0036142999363206.
[2]	K. Feng, On difference schemes and symplectic geometry, in "Proceedings of the 1984 Beijing symposium Symposium on Differential Geometry & Differential Equations," Beijing, (1985), 42-58.
[3]	K. Feng, H. M. Wu, M. Z. Qin and D. L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comp. Math., 7 (1989), 71-96.
[4]	E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag Berlin Heidelberg, 2002.
[5]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.
[6]	J. L. Hong, R. Scherer and L. J. Wang, Midpoint rule for a linear stochastic oscillator with additive noise, Neural Parallel and Scientific Computing, 14 (2006), 1-12.
[7]	J. L. Hong, R. Scherer and L. J. Wang, Predictor-corrector methods for a linear stochastic oscillator with additive noise, Mathematical and Computer Modeling, 46 (2007), 738-764. doi: 10.1016/j.mcm.2006.12.009.
[8]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag Berlin Heidelberg, 1992.
[9]	J. A. Lázaro-Camí and J. P. Ortega, The stochastic Hamilton-Jacobi equation, Journal of Geometric Machanics, 1 (2009), 295-315. doi: 10.3934/jgm.2009.1.295.
[10]	M. Leok and J. J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numer. Anal., 31 (2011), 1497-1532. doi: 10.1093/imanum/drq027.
[11]	Q. Ma, D. Q. Ding and X. H. Ding, Symplectic conditions and stochastic generating functions of stochastic Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Applied Mathematics and Computation, 219 (2012), 635-643. doi: 10.1016/j.amc.2012.06.053.
[12]	X. Mao, "Stochastic Differential Equations and Their Applications," Chichester: Horwood Pub., 1997.
[13]	G. N. Milstein, "Numerical Integration of Stochastic Differential Equations," Kluwer Academic Publishers, 1995.
[14]	G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Symplectic integration of hamiltonian systems with additive noise, SIAM J. Numer. Anal., 39 (2002), 2066-2088. doi: 10.1137/S0036142901387440.
[15]	G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583-1604. doi: 10.1137/S0036142901395588.
[16]	T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 91 (1986), 1-24. doi: 10.1007/BF02722218.
[17]	T. Misawa, A stochastic Hamilton-Jacobi theory in stochastic hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 99 (1987), 179-199. doi: 10.1007/BF02726581.
[18]	R. D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Science, (1983), NS-30, 2669-2671.
[19]	A. H. Strømmen Melbø and D. J. Higham, Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51 (2004), 89-99. doi: 10.1016/j.apnum.2004.02.003.
[20]	R. de Vogelaere, Methods of integration which preserve the contact transformation property of the hamiltonian equations, Report No. 4, Dept. Math., Univ. of Notre Dame, Notre Dame, Ind., 1956.
[21]	L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems," Ph.D thesis, Karlsruhe Institute of Technology, KIT Scientific Publishing, 2007.
[22]	L. J. Wang, J. L. Hong, R. Scherer and F. S. Bai, Dynamics and variational integrators of stochastic Hamiltonian systems, International Journal of Numerical Analysis and Modeling, 6 (2009), 586-602.

show all references

References:

[1]	K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38 (2000), 1626-1646. doi: 10.1137/S0036142999363206.
[2]	K. Feng, On difference schemes and symplectic geometry, in "Proceedings of the 1984 Beijing symposium Symposium on Differential Geometry & Differential Equations," Beijing, (1985), 42-58.
[3]	K. Feng, H. M. Wu, M. Z. Qin and D. L. Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comp. Math., 7 (1989), 71-96.
[4]	E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag Berlin Heidelberg, 2002.
[5]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.
[6]	J. L. Hong, R. Scherer and L. J. Wang, Midpoint rule for a linear stochastic oscillator with additive noise, Neural Parallel and Scientific Computing, 14 (2006), 1-12.
[7]	J. L. Hong, R. Scherer and L. J. Wang, Predictor-corrector methods for a linear stochastic oscillator with additive noise, Mathematical and Computer Modeling, 46 (2007), 738-764. doi: 10.1016/j.mcm.2006.12.009.
[8]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag Berlin Heidelberg, 1992.
[9]	J. A. Lázaro-Camí and J. P. Ortega, The stochastic Hamilton-Jacobi equation, Journal of Geometric Machanics, 1 (2009), 295-315. doi: 10.3934/jgm.2009.1.295.
[10]	M. Leok and J. J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numer. Anal., 31 (2011), 1497-1532. doi: 10.1093/imanum/drq027.
[11]	Q. Ma, D. Q. Ding and X. H. Ding, Symplectic conditions and stochastic generating functions of stochastic Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Applied Mathematics and Computation, 219 (2012), 635-643. doi: 10.1016/j.amc.2012.06.053.
[12]	X. Mao, "Stochastic Differential Equations and Their Applications," Chichester: Horwood Pub., 1997.
[13]	G. N. Milstein, "Numerical Integration of Stochastic Differential Equations," Kluwer Academic Publishers, 1995.
[14]	G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Symplectic integration of hamiltonian systems with additive noise, SIAM J. Numer. Anal., 39 (2002), 2066-2088. doi: 10.1137/S0036142901387440.
[15]	G. N. Milstein, Y. M. Repin and M. V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), 1583-1604. doi: 10.1137/S0036142901395588.
[16]	T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 91 (1986), 1-24. doi: 10.1007/BF02722218.
[17]	T. Misawa, A stochastic Hamilton-Jacobi theory in stochastic hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 99 (1987), 179-199. doi: 10.1007/BF02726581.
[18]	R. D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Science, (1983), NS-30, 2669-2671.
[19]	A. H. Strømmen Melbø and D. J. Higham, Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51 (2004), 89-99. doi: 10.1016/j.apnum.2004.02.003.
[20]	R. de Vogelaere, Methods of integration which preserve the contact transformation property of the hamiltonian equations, Report No. 4, Dept. Math., Univ. of Notre Dame, Notre Dame, Ind., 1956.
[21]	L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems," Ph.D thesis, Karlsruhe Institute of Technology, KIT Scientific Publishing, 2007.
[22]	L. J. Wang, J. L. Hong, R. Scherer and F. S. Bai, Dynamics and variational integrators of stochastic Hamiltonian systems, International Journal of Numerical Analysis and Modeling, 6 (2009), 586-602.

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