Splitting integrators for stochastic Lie–Poisson systems

Bréhier, Charles-Edouard; Cohen, David; Jahnke, Tobias

doi:10.1090/mcom/3829

Splitting integrators for stochastic Lie–Poisson systems
HTML articles powered by AMS MathViewer

by Charles-Edouard Bréhier, David Cohen and Tobias Jahnke HTML | PDF

Math. Comp. 92 (2023), 2167-2216 Request permission

Abstract:

We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie–Poisson systems, namely: stochastically perturbed Maxwell–Bloch, rigid body and sine–Euler equations.

References

Assyr Abdulle, David Cohen, Gilles Vilmart, and Konstantinos C. Zygalakis, High weak order methods for stochastic differential equations based on modified equations, SIAM J. Sci. Comput. 34 (2012), no. 3, A1800–A1823. MR 2970274, DOI 10.1137/110846609
A. Alamo and J. M. Sanz-Serna, A technique for studying strong and weak local errors of splitting stochastic integrators, SIAM J. Numer. Anal. 54 (2016), no. 6, 3239–3257. MR 3570281, DOI 10.1137/16M1058765
A. Alamo and J. M. Sanz-Serna, Word combinatorics for stochastic differential equations: splitting integrators, Commun. Pure Appl. Anal. 18 (2019), no. 4, 2163–2195. MR 3927434, DOI 10.3934/cpaa.2019097
Sverre Anmarkrud and Anne Kværnø, Order conditions for stochastic Runge-Kutta methods preserving quadratic invariants of Stratonovich SDEs, J. Comput. Appl. Math. 316 (2017), 40–46. MR 3588726, DOI 10.1016/j.cam.2016.08.042
Cristina A. Anton, Yau Shu Wong, and Jian Deng, Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions, Int. J. Numer. Anal. Model. 11 (2014), no. 3, 427–451. MR 3218332
Alexis Arnaudon, Alex L. De Castro, and Darryl D. Holm, Noise and dissipation in rigid body motion, Stochastic geometric mechanics, Springer Proc. Math. Stat., vol. 202, Springer, Cham, 2017, pp. 1–12. MR 3747641, DOI 10.1007/978-3-319-63453-1_{1}
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Jean-Michel Bismut, Mécanique aléatoire, Lecture Notes in Mathematics, vol. 866, Springer-Verlag, Berlin-New York, 1981 (French). With an English summary. MR 629977, DOI 10.1007/BFb0088591
Sergio Blanes and Fernando Casas, A concise introduction to geometric numerical integration, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3642447
Nawaf Bou-Rabee and Houman Owhadi, Stochastic variational integrators, IMA J. Numer. Anal. 29 (2009), no. 2, 421–443. MR 2491434, DOI 10.1093/imanum/drn018
Charles-Edouard Bréhier and Shmuel Rakotonirina-Ricquebourg, On asymptotic preserving schemes for a class of stochastic differential equations in averaging and diffusion approximation regimes, Multiscale Model. Simul. 20 (2022), no. 1, 118–163. MR 4376295, DOI 10.1137/20M1379836
Kevin Burrage and Pamela M. Burrage, Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236 (2012), no. 16, 3920–3930. MR 2926251, DOI 10.1016/j.cam.2012.03.007
Sandra Cerrai, Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, vol. 1762, Springer-Verlag, Berlin, 2001. A probabilistic approach. MR 1840644, DOI 10.1007/b80743
Chuchu Chen, David Cohen, Raffaele D’Ambrosio, and Annika Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46 (2020), no. 2, Paper No. 27, 22. MR 4077238, DOI 10.1007/s10444-020-09771-5
Chuchu Chen, Jialing Hong, Diancong Jin, and Liying Sun, Asymptotically-preserving large deviations principles by stochastic symplectic methods for a linear stochastic oscillator, SIAM J. Numer. Anal. 59 (2021), no. 1, 32–59. MR 4194321, DOI 10.1137/19M1306919
David Cohen, Kristian Debrabant, and Andreas Rößler, High order numerical integrators for single integrand Stratonovich SDEs, Appl. Numer. Math. 158 (2020), 264–270. MR 4137619, DOI 10.1016/j.apnum.2020.08.002
David Cohen and Guillaume Dujardin, Energy-preserving integrators for stochastic Poisson systems, Commun. Math. Sci. 12 (2014), no. 8, 1523–1539. MR 3210739, DOI 10.4310/CMS.2014.v12.n8.a7
David Cohen and Gilles Vilmart, Drift-preserving numerical integrators for stochastic Poisson systems, Int. J. Comput. Math. 99 (2022), no. 1, 4–20. MR 4375496, DOI 10.1080/00207160.2021.1922679
D. David and D. D. Holm, Multiple Lie-Poisson structures, reductions, and geometric phases for the Maxwell-Bloch travelling wave equations, J. Nonlinear Sci. 2 (1992), no. 2, 241–262. MR 1169593, DOI 10.1007/BF02429857
Jian Deng, Cristina Anton, and Yau Shu Wong, High-order symplectic schemes for stochastic Hamiltonian systems, Commun. Comput. Phys. 16 (2014), no. 1, 169–200. MR 3195572, DOI 10.4208/cicp.311012.191113a
Kenth Engø and Stig Faltinsen, Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy, SIAM J. Numer. Anal. 39 (2001), no. 1, 128–145. MR 1860719, DOI 10.1137/S0036142999364212
Thomas C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal. 10 (1986), no. 12, 1411–1419. MR 869549, DOI 10.1016/0362-546X(86)90111-2
François Gay-Balmaz and Darryl D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlinear Sci. 28 (2018), no. 3, 873–904. MR 3800250, DOI 10.1007/s00332-017-9431-0
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
Minggang Han, Qiang Ma, and Xiaohua Ding, High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise, Appl. Math. Comput. 346 (2019), 575–593. MR 3873562, DOI 10.1016/j.amc.2018.10.041
Darryl D. Holm and Tomasz M. Tyranowski, Stochastic discrete Hamiltonian variational integrators, BIT 58 (2018), no. 4, 1009–1048. MR 3882980, DOI 10.1007/s10543-018-0720-2
Jialin Hong, Jialin Ruan, Liying Sun, and Lijin Wang, Structure-preserving numerical methods for stochastic Poisson systems, Commun. Comput. Phys. 29 (2021), no. 3, 802–830. MR 4203112, DOI 10.4208/cicp.oa-2019-0084
Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374, DOI 10.1007/978-3-662-12616-5
Joan-Andreu Lázaro-Camí and Juan-Pablo Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys. 61 (2008), no. 1, 65–122. MR 2408499, DOI 10.1016/S0034-4877(08)80003-1
Joan-Andreu Lázaro-Camí and Juan-Pablo Ortega, Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, Stoch. Dyn. 9 (2009), no. 1, 1–46. MR 2502472, DOI 10.1142/S0219493709002531
Benedict Leimkuhler and Sebastian Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. MR 2132573
Xiuyan Li, Chiping Zhang, Qiang Ma, and Xiaohua Ding, Arbitrary high-order EQUIP methods for stochastic canonical Hamiltonian systems, Taiwanese J. Math. 23 (2019), no. 3, 703–725. MR 3952248, DOI 10.11650/tjm/180803
Ming Liao and Longmin Wang, Motion of a rigid body under random perturbation, Electron. Comm. Probab. 10 (2005), 235–243. MR 2198598, DOI 10.1214/ECP.v10-1163
Qiang Ma, Deqiong Ding, and Xiaohua Ding, Symplectic conditions and stochastic generating functions of stochastic Runge-Kutta methods for stochastic Hamiltonian systems with multiplicative noise, Appl. Math. Comput. 219 (2012), no. 2, 635–643. MR 2956993, DOI 10.1016/j.amc.2012.06.053
Simon J. A. Malham and Anke Wiese, Stochastic Lie group integrators, SIAM J. Sci. Comput. 30 (2008), no. 2, 597–617. MR 2385877, DOI 10.1137/060666743
Robert I. McLachlan, Explicit Lie-Poisson integration and the Euler equations, Phys. Rev. Lett. 71 (1993), no. 19, 3043–3046. MR 1246065, DOI 10.1103/PhysRevLett.71.3043
Robert I. McLachlan and G. Reinout W. Quispel, Splitting methods, Acta Numer. 11 (2002), 341–434. MR 2009376, DOI 10.1017/S0962492902000053
G. N. Milstein, Yu. M. Repin, and M. V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal. 40 (2002), no. 4, 1583–1604. MR 1951908, DOI 10.1137/S0036142901395588
G. N. Milstein, Yu. M. Repin, and M. V. Tretyakov, Symplectic integration of Hamiltonian systems with additive noise, SIAM J. Numer. Anal. 39 (2002), no. 6, 2066–2088. MR 1897950, DOI 10.1137/S0036142901387440
G. N. Milstein and M. V. Tretyakov, Stochastic numerics for mathematical physics, Scientific Computation, Springer-Verlag, Berlin, 2004. MR 2069903, DOI 10.1007/978-3-662-10063-9
Tetsuya Misawa, Conserved quantities and symmetry for stochastic dynamical systems, Phys. Lett. A 195 (1994), no. 3-4, 185–189. MR 1306235, DOI 10.1016/0375-9601(94)90150-3
Tetsuya Misawa, Conserved quantities and symmetries related to stochastic dynamical systems, Ann. Inst. Statist. Math. 51 (1999), no. 4, 779–802. MR 1735312, DOI 10.1023/A:1004095516648
T. Misawa, Symplectic integrators to stochastic Hamiltonian dynamical systems derived from composition methods, Math. Probl. Eng. (2010), Art. ID 384937, 12.
Syoiti Ninomiya and Nicolas Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), no. 1-2, 107–121. MR 2409419, DOI 10.1080/13504860701413958
Grigorios A. Pavliotis and Andrew M. Stuart, Multiscale methods, Texts in Applied Mathematics, vol. 53, Springer, New York, 2008. Averaging and homogenization. MR 2382139
Mircea Puta, Lie-Trotter formula and Poisson dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 3, 555–559. Nonlinearity, bifurcation and chaos: the doors to the future, Part II (Dobieszków, 1996). MR 1702129, DOI 10.1142/S0218127499000390
Ryszard Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl. 108 (2003), no. 1, 93–107. MR 2008602, DOI 10.1016/S0304-4149(03)00090-5
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017, DOI 10.1007/978-1-4899-3093-4
Liying Sun and Lijin Wang, Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems, J. Comput. Appl. Math. 311 (2017), 439–456. MR 3552716, DOI 10.1016/j.cam.2016.08.011
Denis Talay and Luciano Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991). MR 1091544, DOI 10.1080/07362999008809220
Tomasz M. Tyranowski, Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations, Proc. A. 477 (2021), no. 2252, Paper No. 20210167, 23. MR 4312508, DOI 10.1098/rspa.2021.0167
J. Walter, O. Gonzalez, and J. H. Maddocks, On the stochastic modeling of rigid body systems with application to polymer dynamics, Multiscale Model. Simul. 8 (2010), no. 3, 1018–1053. MR 2644322, DOI 10.1137/090765705
Lijin Wang and Jialin Hong, Generating functions for stochastic symplectic methods, Discrete Contin. Dyn. Syst. 34 (2014), no. 3, 1211–1228. MR 3094570, DOI 10.3934/dcds.2014.34.1211
Lijin Wang, Pengjun Wang, and Yanzhao Cao, Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), no. 4, 819–836. MR 4396382, DOI 10.3934/dcdss.2021095
L. J. Wang, Variational integrators and generating functions for stochastic Hamiltonian systems, Ph.D. Thesis, Karlsruhe Institute of Technology, 2007.
Peng Wang, Jialin Hong, and Dongsheng Xu, Construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems, Commun. Comput. Phys. 21 (2017), no. 1, 237–270. MR 3579605, DOI 10.4208/cicp.261014.230616a
V. Zeitlin, Finite-mode analogs of $2$D ideal hydrodynamics: coadjoint orbits and local canonical structure, Phys. D 49 (1991), no. 3, 353–362. MR 1115867, DOI 10.1016/0167-2789(91)90152-Y