American Institute of Mathematical Sciences

Advanced
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 - A, 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.  doi: 10.1137/S0036142999363206.  Google Scholar

[2]

K. Feng, On difference schemes and symplectic geometry,, in, (1985), 42.   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.   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.  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.   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.  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.  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.  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.  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.  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.  doi: 10.1137/S0036142901395588.  Google Scholar

[16]

T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes,, Nuovo Cimento B, 91 (1986), 1.  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.  doi: 10.1007/BF02726581.  Google Scholar

[18]

R. D. Ruth, A canonical integration technique,, IEEE Trans. Nuclear Science, (1983), 2669.   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.  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, (1956).   Google Scholar

[21]

L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems,", Ph.D thesis, (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.   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.  doi: 10.1137/S0036142999363206.  Google Scholar

[2]

K. Feng, On difference schemes and symplectic geometry,, in, (1985), 42.   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.   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.  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.   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.  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.  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.  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.  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.  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.  doi: 10.1137/S0036142901395588.  Google Scholar

[16]

T. Misawa, On stochastic Hamiltonian mechanics for diffusion processes,, Nuovo Cimento B, 91 (1986), 1.  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.  doi: 10.1007/BF02726581.  Google Scholar

[18]

R. D. Ruth, A canonical integration technique,, IEEE Trans. Nuclear Science, (1983), 2669.   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.  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, (1956).   Google Scholar

[21]

L. J. Wang, "Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems,", Ph.D thesis, (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.   Google Scholar

[1]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[2]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[3]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[4]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[5]

Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223
[6]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
[7]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[8]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[9]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[10]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
[11]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[12]

Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493
[13]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : ⅰ-ⅲ. doi: 10.3934/dcdss.201805i
[14]

Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917
[15]

Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231
[16]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317
[17]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[18]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[19]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[20]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences