American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag Berlin Heidelberg, 2002.  Google Scholar

[5]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag Berlin Heidelberg, 1992.  Google Scholar

[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.  Google Scholar

[10]

M. Leok and J. J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numer. Anal., 31 (2011), 1497-1532. doi: 10.1093/imanum/drq027.  Google Scholar

[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.  Google Scholar

[12]

X. Mao, "Stochastic Differential Equations and Their Applications," Chichester: Horwood Pub., 1997.  Google Scholar

[13]

G. N. Milstein, "Numerical Integration of Stochastic Differential Equations," Kluwer Academic Publishers, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 91 (1986), 1-24. doi: 10.1007/BF02722218.  Google Scholar

[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.  Google Scholar

[18]

R. D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Science, (1983), NS-30, 2669-2671. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems," Ph.D thesis, Karlsruhe Institute of Technology, KIT Scientific Publishing, 2007. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," Springer-Verlag Berlin Heidelberg, 2002.  Google Scholar

[5]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag Berlin Heidelberg, 1992.  Google Scholar

[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.  Google Scholar

[10]

M. Leok and J. J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numer. Anal., 31 (2011), 1497-1532. doi: 10.1093/imanum/drq027.  Google Scholar

[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.  Google Scholar

[12]

X. Mao, "Stochastic Differential Equations and Their Applications," Chichester: Horwood Pub., 1997.  Google Scholar

[13]

G. N. Milstein, "Numerical Integration of Stochastic Differential Equations," Kluwer Academic Publishers, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes, Nuovo Cimento B, 91 (1986), 1-24. doi: 10.1007/BF02722218.  Google Scholar

[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.  Google Scholar

[18]

R. D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Science, (1983), NS-30, 2669-2671. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems," Ph.D thesis, Karlsruhe Institute of Technology, KIT Scientific Publishing, 2007. Google Scholar

[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.  Google Scholar

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