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July  2019, 18(4): 2163-2195. doi: 10.3934/cpaa.2019097

Word combinatorics for stochastic differential equations: Splitting integrators

1. 

Departamento de Matemática Aplicada e IMUVA, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

2. 

Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Avenida de la Universidad 30, E-28911 Leganés (Madrid), Spain

* Corresponding author

Received  April 2018 Revised  November 2018 Published  January 2019

Fund Project: J. M. S. has been supported by project MTM2016-77660-P(AEI/FEDER, UE), MINECO (Spain)

We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.

Citation: A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097
References:
[1]

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 (2106), 3239-3257. doi: 10.1137/16M1058765. Google Scholar

[2]

F. Baudoin, Diffusion Processes and Stochastic Calculus, European Mathematical Society, Textbooks in Mathematics Vol. 16, 2014. doi: 10.4171/133. Google Scholar

[3]

S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Maths., 54 (2005), 23-37. doi: 10.1016/j.apnum.2004.10.005. Google Scholar

[4] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, CRC Press, Boca Raton, 2016. Google Scholar
[5]

S. BlanesF. CasasA. FarrésJ. LaskarJ. Makazaga and A. Murua, New families of symplectic splitting methods for numerical integration in dynamical astronomy, Appl. Numer. Math., 68 (2013), 58-72. doi: 10.1016/j.apnum.2013.01.003. Google Scholar

[6]

S. BlanesF. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. SeMA, 45 (2008), 89-145. Google Scholar

[7]

Ch. Brouder, Trees, renormalization and differential equations, BIT Numerical Mathematics, 44 (2004), 425-438. doi: 10.1023/B:BITN.0000046809.66837.cc. Google Scholar

[8]

J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963), 185-201. Google Scholar

[9]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 3$^{rd}$ edition, John Wiley & Sons Ltd., Chichester, 2016. doi: 10.1002/9781119121534. Google Scholar

[10]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, in Chaotic Numerics (eds. P. E. Kloeden and K. J. Palmer), Contemporary Mathematics, Vol. 172, American Mathematical Society, Providence, (1994), 63–74. doi: 10.1090/conm/172/01798. Google Scholar

[11]

M. P. Calvo and J. M. Sanz-Serna, Canonical B-series, Numer. Math., 67 (1994), 161-175. doi: 10.1007/s002110050022. Google Scholar

[12]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅰ: B-series, Found. Comput. Math., 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0. Google Scholar

[13]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅱ: the quasi-periodic case, Found. Comput. Math., 12 (2012), 471-508. doi: 10.1007/s10208-012-9118-8. Google Scholar

[14]

P. ChartierA. Murua and J. M. Sanz-Serna, A formal series approach to averaging: exponentially small error estimates, DCDS A, 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009. Google Scholar

[15]

P. ChartierA. Murua and and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found. Comput. Math., 15 (2015), 591-612. doi: 10.1007/s10208-013-9175-7. Google Scholar

[16]

K. T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math., 65 (1957), 163-178. doi: 10.2307/1969671. Google Scholar

[17]

F. Fauvet and F. Menous, Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra, Ann. Sci. Ec. Norm. Sup., 50 (2017), 39-83. doi: 10.24033/asens.2315. Google Scholar

[18]

K. Ebrahimi-FardA. LundervoldS. J. A. MalhamH. Munthe-Kaas and A. Wiese, Algebraic structure of stochastic expansions and efficient simulation, Proc. R. Soc. A, 468 (2012), 2361-2382. doi: 10.1098/rspa.2012.0024. Google Scholar

[19]

M. Fliess, Fonctionnelles causales non-linéaires et indeterminées noncommutatives, Bull. Soc. Math. France, 109 (1981), 3-40. Google Scholar

[20]

J. G. Gaines, The algebra of iterated stochastic integrals, Stochastics, 49 (1994), 169-179. doi: 10.1080/17442509408833918. Google Scholar

[21]

E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration, 2$^{nd}$ edition, Springer, Berlin, 2006. Google Scholar

[22]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1993. Google Scholar

[23]

E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing, 13 (1974), 1-15. Google Scholar

[24]

M. E. Hoffman, Quasi-shuffle products, J. Algbr. Comb., 11 (2000), 49-68. doi: 10.1023/A:1008791603281. Google Scholar

[25]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. Google Scholar

[26]

B. Leimkuhler and C. Matthews, Rational construction of stochastic numerical methods for molecular sampling, App. Math. Res. Express, 2013 (2013), 34-56. Google Scholar

[27]

B. Leimkuhler and C. Matthews, Robust and efficient configurational molecular sampling via Langevin Dynamics, J. Chem. Phys., 138 (2013), 174102. Google Scholar

[28]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. Google Scholar

[29]

R. H. Merson, An operational method for the study of integration processes, in Proceedings of the Symposium on Data Processing, Weapons Researcch Establishement, Salisbury, Australia, (1957), 110.1–110.25.Google Scholar

[30]

G. N. Miltstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004. doi: 10.1007/978-3-662-10063-9. Google Scholar

[31]

A. Murua, The Hopf algebra of rooted trees, free Lie algebras and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0. Google Scholar

[32]

A. Murua and J. M. Sanz-Serna, Order conditions for numerical integrators obtained by composing simpler integrators, Phil. Trans. R. Soc. Lond. A, 357 (1999), 1079-1100. doi: 10.1098/rsta.1999.0365. Google Scholar

[33]

A. Murua and J. M. Sanz-Serna, Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Anal.-Theor., 138 (2016), 326-345. doi: 10.1016/j.na.2015.10.013. Google Scholar

[34]

A. Murua and J. M. Sanz-Serna, Vibrational resonance: a study with high-order word-series averaging, Applied Mathematics and Nonlinear Sciences, 1 (2016), 239-246. Google Scholar

[35]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found. Comput. Math., 17 (2017), 675-712. doi: 10.1007/s10208-015-9295-3. Google Scholar

[36]

A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms, in Discrete Mechanics, Geometric Integration and Lie-Butcher Series (eds. K. Ebrahimi-Fard and M. Barbero Liñan), Springer, Berlin, (2018), 115–137. doi: 10.1007/s10208-010-9074-0. Google Scholar

[37]

A. Murua and J. M. Sanz-Serna, Hopf algebra techniques to handle dynamical systems and numerical integrators, in Computation and Combinatorics in Dynamics, Stochastics and Control, The Abel Symposium, Rosendal, Norway, August 2016 (eds. E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard and H. Z. Munthe-Kaas), Springer, (2019), to appear.Google Scholar

[38]

G. A. Pavliotis, Stochastic Processes and Applications, Springer, New York, 2014. doi: 10.1007/978-1-4939-1323-7. Google Scholar

[39]

E. Platen and W. Wagner, On a Taylor formula for a class of Ito processes, Probab. Math. Statist., 3 (1982), 37-51. Google Scholar

[40]

R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math., 68 (1958), 210-220. doi: 10.2307/1970243. Google Scholar

[41] C. Reutenauer, Free Lie Algebras, Clarendon Press, Oxford, 1993. Google Scholar
[42]

J. M. Sanz-Serna, Geometric integration, in State of the Art in Numerical Analysis (eds. I. S. Duff and G. A. Watson), Clarendon Press, Oxford, (1997), 121–143. Google Scholar

[43]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994. Google Scholar

[44]

J. M. Sanz-Serna and A. Murua, Formal series and numerical integrators: some history and some new techniques, in Proceedings of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015) (eds. Lei-Guo and Zhiming-Ma), Higher Education Press, Beijing, (2015), 311–331. Google Scholar

show all references

References:
[1]

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 (2106), 3239-3257. doi: 10.1137/16M1058765. Google Scholar

[2]

F. Baudoin, Diffusion Processes and Stochastic Calculus, European Mathematical Society, Textbooks in Mathematics Vol. 16, 2014. doi: 10.4171/133. Google Scholar

[3]

S. Blanes and F. Casas, On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Maths., 54 (2005), 23-37. doi: 10.1016/j.apnum.2004.10.005. Google Scholar

[4] S. Blanes and F. Casas, A Concise Introduction to Geometric Numerical Integration, CRC Press, Boca Raton, 2016. Google Scholar
[5]

S. BlanesF. CasasA. FarrésJ. LaskarJ. Makazaga and A. Murua, New families of symplectic splitting methods for numerical integration in dynamical astronomy, Appl. Numer. Math., 68 (2013), 58-72. doi: 10.1016/j.apnum.2013.01.003. Google Scholar

[6]

S. BlanesF. Casas and A. Murua, Splitting and composition methods in the numerical integration of differential equations, Bol. Soc. Esp. Mat. Apl. SeMA, 45 (2008), 89-145. Google Scholar

[7]

Ch. Brouder, Trees, renormalization and differential equations, BIT Numerical Mathematics, 44 (2004), 425-438. doi: 10.1023/B:BITN.0000046809.66837.cc. Google Scholar

[8]

J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963), 185-201. Google Scholar

[9]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 3$^{rd}$ edition, John Wiley & Sons Ltd., Chichester, 2016. doi: 10.1002/9781119121534. Google Scholar

[10]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, in Chaotic Numerics (eds. P. E. Kloeden and K. J. Palmer), Contemporary Mathematics, Vol. 172, American Mathematical Society, Providence, (1994), 63–74. doi: 10.1090/conm/172/01798. Google Scholar

[11]

M. P. Calvo and J. M. Sanz-Serna, Canonical B-series, Numer. Math., 67 (1994), 161-175. doi: 10.1007/s002110050022. Google Scholar

[12]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅰ: B-series, Found. Comput. Math., 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0. Google Scholar

[13]

P. ChartierA. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅱ: the quasi-periodic case, Found. Comput. Math., 12 (2012), 471-508. doi: 10.1007/s10208-012-9118-8. Google Scholar

[14]

P. ChartierA. Murua and J. M. Sanz-Serna, A formal series approach to averaging: exponentially small error estimates, DCDS A, 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009. Google Scholar

[15]

P. ChartierA. Murua and and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found. Comput. Math., 15 (2015), 591-612. doi: 10.1007/s10208-013-9175-7. Google Scholar

[16]

K. T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math., 65 (1957), 163-178. doi: 10.2307/1969671. Google Scholar

[17]

F. Fauvet and F. Menous, Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra, Ann. Sci. Ec. Norm. Sup., 50 (2017), 39-83. doi: 10.24033/asens.2315. Google Scholar

[18]

K. Ebrahimi-FardA. LundervoldS. J. A. MalhamH. Munthe-Kaas and A. Wiese, Algebraic structure of stochastic expansions and efficient simulation, Proc. R. Soc. A, 468 (2012), 2361-2382. doi: 10.1098/rspa.2012.0024. Google Scholar

[19]

M. Fliess, Fonctionnelles causales non-linéaires et indeterminées noncommutatives, Bull. Soc. Math. France, 109 (1981), 3-40. Google Scholar

[20]

J. G. Gaines, The algebra of iterated stochastic integrals, Stochastics, 49 (1994), 169-179. doi: 10.1080/17442509408833918. Google Scholar

[21]

E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration, 2$^{nd}$ edition, Springer, Berlin, 2006. Google Scholar

[22]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1993. Google Scholar

[23]

E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing, 13 (1974), 1-15. Google Scholar

[24]

M. E. Hoffman, Quasi-shuffle products, J. Algbr. Comb., 11 (2000), 49-68. doi: 10.1023/A:1008791603281. Google Scholar

[25]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. Google Scholar

[26]

B. Leimkuhler and C. Matthews, Rational construction of stochastic numerical methods for molecular sampling, App. Math. Res. Express, 2013 (2013), 34-56. Google Scholar

[27]

B. Leimkuhler and C. Matthews, Robust and efficient configurational molecular sampling via Langevin Dynamics, J. Chem. Phys., 138 (2013), 174102. Google Scholar

[28]

R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. Google Scholar

[29]

R. H. Merson, An operational method for the study of integration processes, in Proceedings of the Symposium on Data Processing, Weapons Researcch Establishement, Salisbury, Australia, (1957), 110.1–110.25.Google Scholar

[30]

G. N. Miltstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004. doi: 10.1007/978-3-662-10063-9. Google Scholar

[31]

A. Murua, The Hopf algebra of rooted trees, free Lie algebras and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0. Google Scholar

[32]

A. Murua and J. M. Sanz-Serna, Order conditions for numerical integrators obtained by composing simpler integrators, Phil. Trans. R. Soc. Lond. A, 357 (1999), 1079-1100. doi: 10.1098/rsta.1999.0365. Google Scholar

[33]

A. Murua and J. M. Sanz-Serna, Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Anal.-Theor., 138 (2016), 326-345. doi: 10.1016/j.na.2015.10.013. Google Scholar

[34]

A. Murua and J. M. Sanz-Serna, Vibrational resonance: a study with high-order word-series averaging, Applied Mathematics and Nonlinear Sciences, 1 (2016), 239-246. Google Scholar

[35]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found. Comput. Math., 17 (2017), 675-712. doi: 10.1007/s10208-015-9295-3. Google Scholar

[36]

A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms, in Discrete Mechanics, Geometric Integration and Lie-Butcher Series (eds. K. Ebrahimi-Fard and M. Barbero Liñan), Springer, Berlin, (2018), 115–137. doi: 10.1007/s10208-010-9074-0. Google Scholar

[37]

A. Murua and J. M. Sanz-Serna, Hopf algebra techniques to handle dynamical systems and numerical integrators, in Computation and Combinatorics in Dynamics, Stochastics and Control, The Abel Symposium, Rosendal, Norway, August 2016 (eds. E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard and H. Z. Munthe-Kaas), Springer, (2019), to appear.Google Scholar

[38]

G. A. Pavliotis, Stochastic Processes and Applications, Springer, New York, 2014. doi: 10.1007/978-1-4939-1323-7. Google Scholar

[39]

E. Platen and W. Wagner, On a Taylor formula for a class of Ito processes, Probab. Math. Statist., 3 (1982), 37-51. Google Scholar

[40]

R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math., 68 (1958), 210-220. doi: 10.2307/1970243. Google Scholar

[41] C. Reutenauer, Free Lie Algebras, Clarendon Press, Oxford, 1993. Google Scholar
[42]

J. M. Sanz-Serna, Geometric integration, in State of the Art in Numerical Analysis (eds. I. S. Duff and G. A. Watson), Clarendon Press, Oxford, (1997), 121–143. Google Scholar

[43]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994. Google Scholar

[44]

J. M. Sanz-Serna and A. Murua, Formal series and numerical integrators: some history and some new techniques, in Proceedings of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015) (eds. Lei-Guo and Zhiming-Ma), Higher Education Press, Beijing, (2015), 311–331. Google Scholar

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