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Split S-ROCK Methods for High-Dimensional Stochastic Differential Equations

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Abstract

We propose explicit stochastic Runge–Kutta (RK) methods for high-dimensional Itô stochastic differential equations. By providing a linear error analysis and utilizing a Strang splitting-type approach, we construct them on the basis of orthogonal Runge–Kutta–Chebyshev methods of order 2. Our methods are of weak order 2 and have high computational accuracy for relatively large time-step size, as well as good stability properties. In addition, we take stochastic exponential RK methods of weak order 2 as competitors, and deal with implementation issues on Krylov subspace projection techniques for them. We carry out numerical experiments on a variety of linear and nonlinear problems to check the computational performance of the methods. As a result, it is shown that the proposed methods can be very effective on high-dimensional problems whose drift term has eigenvalues lying near the negative real axis and whose diffusion term does not have very large noise.

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Data Availability

Data and program codes are available in https://github.com/yosh-komori/supplementary_info_files_2023.

Notes

  1. https://www.epfl.ch/labs/anmc/software/.

  2. For an implementation of the methods, we utilize the parameter values in a Fortran code, rectp.f, obtained from http://anmc.epfl.ch/Pdf/srock2.zip.

References

  1. Abdulle, A., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30(2), 997–1014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdulle, A., Li, T.: S-ROCK methods for stiff Itô SDEs. Commun. Math. Sci. 6(4), 845–868 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abdulle, A., Medovikov, A.A.: Second order Chebyshev methods based on orthogonal polynomials. Numer. Math. 90, 1–18 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abdulle, A., Vilmart, G., Zygalakis, K.C.: Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 35(4), A1792–A1814 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abdulle, A., Almuslimani, I., Vilmart, G.: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 6(2), 937–964 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Al-Mohy, A.H., Higham, N.J.: A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. 31(3), 970–989 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold, L.: Stochastic Differential Equations: Theory and Applications. John Wiley & Sons, New York (1974)

    MATH  Google Scholar 

  8. Brzeźniak, Z., Goldys, B., Neklyudov, M.: Multidimensional stochastic Burgers equation. SIAM J. Math. Anal. 46(1), 871–889 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cohen, D.: On the numerical discretisation of stochastic oscillators. Math. Comput. Simul. 82(8), 1478–1495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen, D., Sigg, M.: Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations. Numer. Math. 121(1), 1–29 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Debrabant, K., Kværnø, A., Mattsson, N.C.: Runge-Kutta Lawson schemes for stochastic differential equations. BIT Numer. Math. 61(2), 381–409 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Debrabant, K., Kværnø, A., Mattsson, N.C.: Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants. BIT Numer. Math. 62(4), 1121–1147 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Anal. 9, 1–25 (1998)

    Article  MathSciNet  Google Scholar 

  14. Erdoğan, U., Lord, G.J.: A new class of exponential integrators for SDEs with multiplicative noise. IMA J. Appl. Math. 39(2), 820–846 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ibáñez, J., Alonso, J.M., Alonso-Jordá, P., Defez, E., Sastre, J.: Two Taylor algorithms for computing the action of the matrix exponential on a vector. Algorithms 15(2), 48 (2022)

    Article  Google Scholar 

  19. Kamm, K., Pagliarani, S., Pascucci, A.: Numerical solution of kinetic SPDEs via stochastic Magnus expansion. Math. Comput. Simul. 207, 189–208 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin (1999). (Corrected Third Printing)

    MATH  Google Scholar 

  21. Komori, Y., Burrage, K.: Weak second order S-ROCK methods for Stratonovich stochastic differential equations. J. Comput. Appl. Math. 236(11), 2895–2908 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Komori, Y., Burrage, K.: A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems. BIT 54(4), 1067–1085 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Komori, Y., Cohen, D., Burrage, K.: Weak second order explicit exponential Runge-Kutta methods for stochastic differential equations. SIAM J. Sci. Comput. 39(6), A2857–A2878 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)

    Article  MATH  Google Scholar 

  25. Milstein, G.N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30(4), 750–766 (1986)

    Article  MathSciNet  Google Scholar 

  26. Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Springer-Verlag, Berlin (2004)

    Book  MATH  Google Scholar 

  27. Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)

    Article  MATH  Google Scholar 

  29. Tocino, A., Senosiain, M.J.: Mean-square stability analysis of numerical schemes for stochastic differential systems. J. Comput. Appl. Math. 236(10), 2660–2672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. van der Houwen, P.J., Sommeijer, B.P.: On the internal stability of explicit, \(m\)-stage Runge-Kutta methods for large \(m\)-values. ZAMM Z. Angew. Math. Mech. 60, 479–485 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, G., Burrage, K., Burrage, P., Ding, X.: How to choose an appropriate initial condition to simulate stochastic differential equations stably. In: AIP Conference Proceedings (to appear)

  32. Yang, G., Burrage, K., Komori, Y., Burrage, P.M., Ding, X.: A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations. Numer. Algorithms 88(4), 1641–1665 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank referees, Professor Chi-Wang Shu and Professor David Cohen for their comments which helped to improve the earlier versions of this paper.

Funding

This work was partially supported by JSPS Grant-in-Aid for Scientific Research 17K05369.

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Authors and Affiliations

  1. Department of Physics and Information Technology, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, 820-8502, Japan

    Yoshio Komori

  2. School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia

    Kevin Burrage

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  1. Yoshio Komori
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  2. Kevin Burrage
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Correspondence to Yoshio Komori.

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Komori, Y., Burrage, K. Split S-ROCK Methods for High-Dimensional Stochastic Differential Equations. J Sci Comput 97, 62 (2023). https://doi.org/10.1007/s10915-023-02354-8

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  • DOI: https://doi.org/10.1007/s10915-023-02354-8

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