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Explicit Methods for Stiff Stochastic Differential Equations

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Numerical Analysis of Multiscale Computations

Part of the book series: Lecture Notes in Computational Science and Engineering ((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. A. Abdulle, On roots and error constant of optimal stability polynomials, BIT 40 (2000), no. 1, 177–182.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Abdulle and A.A. Medovikov, Second order Chebyshev methods based on orthogonal polynomials, Numer. Math., 90 (2001), no. 1, 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput., 23 (2002), no. 6, 2041–2054.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  5. A. Abdulle Multiscale methods for advection-diffusion problems, Discrete Contin. Dyn. Syst. (2005), suppl., 11–21.

    Google Scholar 

  6. A. Abdulle and S. Cirilli, Stabilized methods for stiff stochastic systems, C. R. Acad. Sci. Paris, 345 (2007), no. 10, 593–598.

    MathSciNet  MATH  Google Scholar 

  7. A. Abdulle and S. Cirilli, S-ROCK methods for stiff stochastic problems, SIAM J. Sci. Comput., 30 (2008), no. 2, 997–1014.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Abdulle and T. Li, S-ROCK methods for stiff Itô SDEs, Commun. Math. Sci. 6 (2008), no. 4, 845–868.

    MathSciNet  MATH  Google Scholar 

  9. A. Abdulle, Y. Hu and T. Li, Chebyshev methods with discrete noise: the tau-ROCK methods, J. Comput. Math. 28 (2010), no. 2, 195–217

    MathSciNet  MATH  Google Scholar 

  10. L. Arnold, Stochastic differential equation, Theory and Application, Wiley, 1974.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  12. P.M. Burrage, Runge-Kutta methods for stochastic differential equations. PhD Thesis, University of Queensland, Brisbane, Australia, 1999.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. L. Chow, Stochastic partial differential equations, Chapman and Hall/CRC, 2007.

    Google Scholar 

  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. 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. D.T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58 (2007), 35–55.

    Article  Google Scholar 

  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. E. Hairer and G. Wanner, Intégration numérique des équations différentielles raides, Techniques de l’ingénieur AF 653, 2007.

    Google Scholar 

  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. R.Z. Has’minskiǐ, Stochastic stability of differential equations. Sijthoff & Noordhoff, Groningen, The Netherlands, 1980.

    Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  24. D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43 (2001), 525–546.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  26. N.G. van Kampen, Stochastic processes in physics and chemistry, 3rd ed., North-Holland Personal Library, Elsevier, 2007.

    Google Scholar 

  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.

    Article  Google Scholar 

  28. P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics 23, Springer-Verlag, Berlin, 1992.

    Google Scholar 

  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. 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. T. Li, Analysis of explicit tau-leaping schemes for simulating chemically reacting systems, SIAM Multiscale Model. Simul., 6 (2007), no. 2, 417–436.

    Article  MATH  Google Scholar 

  32. G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo, 4 (1955), 48–90.

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Oksendal, Stochastic differential equations, Sixth edition, Springer-Verlag, Berlin, 2003.

    Book  Google Scholar 

  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. E. Platen, Zur zeitdiskreten approximation von Itôprozessen, Diss. B. Imath. Akad. des Wiss. der DDR, Berlin, 1984.

    Google Scholar 

  36. W. Rümlin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Math., 19 (1982), no. 3, 604–613.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. E. Vanden-Eijnden, Numerical techniques for multiscale dynamical system with stochastic effects, Commun. Math. Sci., 1 (2003), no. 2, 385–391.

    MathSciNet  MATH  Google Scholar 

  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. D.J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems, Nature Reviews Genetics 10 (2009), 122–133.

    Article  Google Scholar 

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Authors and Affiliations

  1. Section of Mathematics, Swiss Federal Institute of Technology (EPFL), Station 8, CH-1015, Lausanne, Switzerland

    Assyr Abdulle

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  1. Assyr Abdulle
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Correspondence to Assyr Abdulle .

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Editors and Affiliations

  1. , Department of Mathematics, University of Texas, University Station C 1200 1, Austin, 78712-0257, Texas, USA

    Björn Engquist

  2. Dept. Numerical Analysis &, Computer Science (NADA), Royal Institute of Technology (KTH), Lindstedtsvägen 3, Stockholm, 100 44, Sweden

    Olof Runborg

  3. , Department of Mathematics, University of Texas at Austin, University Station C1200 1, Austin, 78712-0257, Texas, USA

    Yen-Hsi R. Tsai

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Abdulle, A. (2012). Explicit Methods for Stiff Stochastic Differential Equations. In: Engquist, B., Runborg, O., Tsai, YH. (eds) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21943-6_1

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