-
Previous Article
Generating functions and volume preserving mappings
- DCDS Home
- This Issue
-
Next Article
The Landau--Kolmogorov inequality revisited
Generating functions for stochastic symplectic methods
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 |
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. |
[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] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
[3] |
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 |
[4] |
Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 |
[5] |
Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 |
[6] |
Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli. Constrained systems, generalized Hamilton-Jacobi actions, and quantization. Journal of Geometric Mechanics, 2022, 14 (2) : 179-272. doi: 10.3934/jgm.2022010 |
[7] |
Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223 |
[8] |
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 |
[9] |
Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 |
[10] |
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 |
[11] |
Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 |
[12] |
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 |
[13] |
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 |
[14] |
Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022061 |
[15] |
Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493 |
[16] |
Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations and Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 |
[17] |
Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i |
[18] |
Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 |
[19] |
Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 |
[20] |
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 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]