Explicit Methods for Stiff Stochastic Differential Equations

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 82)

Abstract

Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the Euler-Maruyama method, face severe stepsize restriction when applied to stiff problems. Fully implicit methods are usually not appropriate for stochastic problems and semi-implicit methods (implicit in the deterministic part) involve the solution of possibly large linear systems at each time-step. In this paper, we present a recent generalization of explicit stabilized methods, known as Chebyshev methods, to stochastic problems. These methods have much better (mean-square) stability properties than standard explicit methods. We discuss the construction of this new class of methods and illustrate their performance on various problems involving stochastic ordinary and partial differential equations.

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References

  1. 1.
    A. Abdulle, On roots and error constant of optimal stability polynomials, BIT 40 (2000), no. 1, 177–182.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A. Abdulle and A.A. Medovikov, Second order Chebyshev methods based on orthogonal polynomials, Numer. Math., 90 (2001), no. 1, 1–18.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput., 23 (2002), no. 6, 2041–2054.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. Abdulle and S. Attinger, Homogenization method for transport of DNA particles in heterogeneous arrays, Multiscale Modelling and Simulation, Lect. Notes Comput. Sci. Eng., 39 (2004), 23–33.MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Abdulle Multiscale methods for advection-diffusion problems, Discrete Contin. Dyn. Syst. (2005), suppl., 11–21.Google Scholar
  6. 6.
    A. Abdulle and S. Cirilli, Stabilized methods for stiff stochastic systems, C. R. Acad. Sci. Paris, 345 (2007), no. 10, 593–598.MathSciNetMATHGoogle Scholar
  7. 7.
    A. Abdulle and S. Cirilli, S-ROCK methods for stiff stochastic problems, SIAM J. Sci. Comput., 30 (2008), no. 2, 997–1014.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Abdulle and T. Li, S-ROCK methods for stiff Itô SDEs, Commun. Math. Sci. 6 (2008), no. 4, 845–868.MathSciNetMATHGoogle Scholar
  9. 9.
    A. Abdulle, Y. Hu and T. Li, Chebyshev methods with discrete noise: the tau-ROCK methods, J. Comput. Math. 28 (2010), no. 2, 195–217MathSciNetMATHGoogle Scholar
  10. 10.
    L. Arnold, Stochastic differential equation, Theory and Application, Wiley, 1974.Google Scholar
  11. 11.
    E. Buckwar and C. Kelly, Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal. 48 (2010), no. 1, 298–321.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    P.M. Burrage, Runge-Kutta methods for stochastic differential equations. PhD Thesis, University of Queensland, Brisbane, Australia, 1999.Google Scholar
  13. 13.
    K. Burrage and P.M. Burrage, General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Eight Conference on the Numerical Treatment of Differential Equations (Alexisbad, 1997), Appl. Numer. Math. 28 (1998), no. 2-4, 161–177.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    P. L. Chow, Stochastic partial differential equations, Chapman and Hall/CRC, 2007.Google Scholar
  15. 15.
    M. Duarte, M. Massota, S. Descombes, C. Tenaudc, T. Dumont, V. Louvet and F. Laurent, New resolution strategy for multi-scale reaction waves using time operator splitting, space adaptive multiresolution and dedicated high order implicit/explicit time integrators, preprint available at hal.archive ouvertes, 2010.Google Scholar
  16. 16.
    W. E, D. Liu, and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math. 58 (2004), no. 11, 1544–1585.Google Scholar
  17. 17.
    D.T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58 (2007), 35–55.CrossRefGoogle Scholar
  18. 18.
    A. Guillou and B. Lago, Domaine de stabilité associé aux formules d’intégration numérique d’équations différentielles à pas séparés et à pas liés. Recherche de formules à grand rayon de stabilité, in Proceedings of the 1er Congr. Assoc. Fran. Calcul (AFCAL), Grenoble, (1960), 43–56.Google Scholar
  19. 19.
    E. Hairer and G. Wanner, Intégration numérique des équations différentielles raides, Techniques de l’ingénieur AF 653, 2007.Google Scholar
  20. 20.
    E. Hairer and G. Wanner, Solving ordinary differential equations II. Stiff and differential-algebraic problems. 2nd. ed., Springer-Verlag, Berlin, 1996.Google Scholar
  21. 21.
    R.Z. Has’minskiǐ, Stochastic stability of differential equations. Sijthoff & Noordhoff, Groningen, The Netherlands, 1980.Google Scholar
  22. 22.
    M. Hauth, J. Gross, W. Strasser and G.F. Buess, Soft tissue simulation based on measured data, Lecture Notes in Comput. Sci., 2878 (2003), 262–270.CrossRefGoogle Scholar
  23. 23.
    D.J. Higham, Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations, SIAM J. Numer Anal., 38 (2000), no. 3, 753–769.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43 (2001), 525–546.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    P.J. van der Houwen and B.P. Sommeijer, On the internal stage Runge-Kutta methods for large m-values, Z. Angew. Math. Mech., 60 (1980), 479–485.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    N.G. van Kampen, Stochastic processes in physics and chemistry, 3rd ed., North-Holland Personal Library, Elsevier, 2007.Google Scholar
  27. 27.
    A.R. Kinjo and S. Takada, Competition between protein folding and aggregation with molecular chaperones in crowded solutions: insight from mesoscopic simulations, Biophysical journal, 85 (2003), 3521–3531.CrossRefGoogle Scholar
  28. 28.
    P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics 23, Springer-Verlag, Berlin, 1992.Google Scholar
  29. 29.
    V.I. Lebedev, How to solve stiff systems of differential equations by explicit methods. CRC Pres, Boca Raton, FL, (1994), 45–80.Google Scholar
  30. 30.
    T. Li, A. Abdulle and Weinan E, Effectiveness of implicit methods for stiff stochastic differential equations, Commun. Comput. Phys., 3 (2008), no. 2, 295–307.Google Scholar
  31. 31.
    T. Li, Analysis of explicit tau-leaping schemes for simulating chemically reacting systems, SIAM Multiscale Model. Simul., 6 (2007), no. 2, 417–436.MATHCrossRefGoogle Scholar
  32. 32.
    G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo, 4 (1955), 48–90.MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    B. Oksendal, Stochastic differential equations, Sixth edition, Springer-Verlag, Berlin, 2003.CrossRefGoogle Scholar
  34. 34.
    E. Platen and N. Bruti-Liberati, Numerical solutions of stochastic differential equations with jumps in finance, Stochastic Modelling and Applied Probability, Vol. 64, Springer-Verlag, Berlin, 2010.Google Scholar
  35. 35.
    E. Platen, Zur zeitdiskreten approximation von Itôprozessen, Diss. B. Imath. Akad. des Wiss. der DDR, Berlin, 1984.Google Scholar
  36. 36.
    W. Rümlin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Math., 19 (1982), no. 3, 604–613.CrossRefGoogle Scholar
  37. 37.
    Y. Saitô and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), no. 6, 2254–2267.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    V. Thomee, Galerkin finite element methods for parabolic problems, 2nd ed, Springer Series in Computational Mathematics, Vol. 25, Springer-Verlag, Berlin, 2006.Google Scholar
  39. 39.
    E. Vanden-Eijnden, Numerical techniques for multiscale dynamical system with stochastic effects, Commun. Math. Sci., 1 (2003), no. 2, 385–391.MathSciNetMATHGoogle Scholar
  40. 40.
    J.B. Walsh, An introduction to stochastic partial differential equations, In: École d’été de Prob. de St-Flour XIV-1984, Lect. Notes in Math. 1180, Springer-Verlag, Berlin, 1986.Google Scholar
  41. 41.
    D.J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems, Nature Reviews Genetics 10 (2009), 122–133.CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Section of MathematicsSwiss Federal Institute of Technology (EPFL)LausanneSwitzerland

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