Advertisement

Stability and Strong Convergence for Spatial Stochastic Kinetics

  • Stefan Engblom
Chapter

Abstract

We review conditions for the well-posedness of models of stochastic jump kinetics. Our focus is on obtaining bounds in the sense of mean square, implying in particular so-called strong convergence. We look especially on problems posed in a spatial setting, formed by merging a local reaction process with a connecting transport mechanism. This type of network jump process occurs naturally in many applications and is an attractive modeling framework, yet is a challenge from the perspective of numerical analysis. Since the stochastic modeling itself is motivated by the presence of nonlinear feedback terms, by small number of participating agents, and by an overall noisy environment, a consistent analysis framework is clearly required. The review summarizes the required mathematical framework and techniques used for obtaining a priori bounds and stability estimates.

Keywords

Well-posedness Continuous-time Markov chain Network jump process Perturbation Rate equation Mean square bounds 

Mathematics Subject Classification (2010)

60J27 60J28 92C42 

References

  1. 1.
    D.F. Anderson, An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM J. Numer. Anal. 50(5), 2237–2258 (2012). doi:10.1137/110849079MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D.F. Anderson, D.J. Higham, Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics. Multiscale Model. Simul. 10(1), 146–179 (2012). doi:10.1137/110840546MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    D.F. Anderson, M. Koyama, Weak error analysis of numerical methods for stochastic models of population processes. Multiscale Model. Simul. 10(4), 1493–1524 (2012). doi:10.1137/110849699MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    D.F. Anderson, A. Ganguly, T.G. Kurtz, Error analysis of tau-leap simulation methods. Ann. Appl. Probab. 21(6), 2226–2262 (2011). doi:10.1214/10-AAP756MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    D. Applebaum, Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 93 (Cambridge University Press, Cambridge, 2004)Google Scholar
  6. 6.
    G. Arampatzis, M. Katsoulakis, P. Plecháč, Parallelization, processor communication and error analysis in lattice kinetic monte carlo. SIAM J. Numer. Anal. 52(3), 1156–1182 (2014). doi:10.1137/120889459MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Bauer, S. Engblom, Sensitivity estimation and inverse problems in spatial stochastic models of chemical kinetics, in Numerical Mathematics and Advanced Applications: ENUMATH 2013, ed. by A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso. Lecture Notes in Computational Science and Engineering, vol. 103 (Springer, Berlin, 2015), pp. 519–527. doi:10.1007/978-3-319-10705-9_51
  8. 8.
    P. Bauer, S. Engblom, S. Widgren. Fast event-based epidemiological simulations on national scales. Int. J. High Perform. Comput. Appl. 30(4), 438–453 (2016). doi:10.1177/1094342016635723CrossRefGoogle Scholar
  9. 9.
    E. Blanc, S. Engblom, A. Hellander, P. Lötstedt, Mesoscopic modeling of stochastic reaction-diffusion kinetics in the subdiffusive regime. Multiscale Model. Simul. 14(2), 668–707 (2016). doi:10.1137/15M1013110MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics, vol. 31 (Springer, New York, 1999)Google Scholar
  11. 11.
    C. Briat, A. Gupta, M. Khammash, A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLoS Comput. Biol. 10(6), e1003669 (2014). doi:10.1371/journal.pcbi.1003669Google Scholar
  12. 12.
    A. Chevallier, S. Engblom, Pathwise error bounds in multiscale variable splitting methods for spatial stochastic kinetics, 2016. Available at https://arxiv.org/abs/1607.00805 Google Scholar
  13. 13.
    M. Dobrzyński, J.V. Rodríguez, J.A. Kaandorp, J.G. Blom, Computational methods for diffusion-influenced biochemical reactions. Bioinformatics 23(15), 1969–1977 (2007). doi:10.1093/bioinformatics/btm278CrossRefGoogle Scholar
  14. 14.
    B. Drawert, S. Engblom, A. Hellander, URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst. Biol. 6(76), 1–17 (2012) doi:10.1186/1752-0509-6-76Google Scholar
  15. 15.
    S. Engblom, Parallel in time simulation of multiscale stochastic chemical kinetics. Multiscale Model. Simul. 8(1), 46–68 (2009). doi:10.1137/080733723MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    S. Engblom (ed.), Student’s Book: Numerical Functional Analysis (Uppsala University, Uppsala, 2014). Available at http://www.it.uu.se/grad/courses/scicomp/NumFunkAnalysis/NFAStudentBook.pdf Google Scholar
  17. 17.
    S. Engblom, On the stability of stochastic jump kinetics. Appl. Math. 5(19), 3217–3239 (2014) doi:10.4236/am.2014.519300CrossRefGoogle Scholar
  18. 18.
    S. Engblom, Strong convergence for split-step methods in stochastic jump kinetics. SIAM J. Numer. Anal. 53(6), 2655–2676 (2015). doi:10.1137/141000841MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    S. Engblom, L. Ferm, A. Hellander, P. Lötstedt, Simulation of stochastic reaction-diffusion processes on unstructured meshes. SIAM J. Sci. Comput. 31(3), 1774–1797 (2009). doi:10.1137/080721388MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1986)Google Scholar
  21. 21.
    D. Fange, J. Elf, Noise-induced Min phenotypes in E. coli. PLoS Comput. Biol. 2(6), 637–648 (2006). doi:10.1371/journal.pcbi.0020080Google Scholar
  22. 22.
    A. Ganguly, D. Altıntan, H. Koeppl, Jump-diffusion approximation of stochastic reaction dynamics: error bounds and algorithms. Multiscale Model. Simul. 13(4), 1390–1419 (2015). doi:10.1137/140983471MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    C.W. Gardiner, Handbook of Stochastic Methods. Springer Series in Synergetics, 3rd edn. (Springer, Berlin, 2004)Google Scholar
  24. 24.
    D.T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434 (1976). doi:10.1016/0021-9991(76)90041-3MathSciNetCrossRefGoogle Scholar
  25. 25.
    E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results. Probab. Theory Relat. Fields 137(1–2), 161–200 (2007). doi:10.1007/s00440-006-0501-8MathSciNetMATHGoogle Scholar
  26. 26.
    D.J. Higham, P.E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 101(1), 101–119 (2005). doi:10.1007/s00211-005-0611-8MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J. Karlsson, R. Tempone, Towards automatic global error control: computable weak error expansion for the tau-leap method. Monte Carlo Methods Appl. 17(3), 233–278 (2011). doi:10.1515/MCMA.2011.011MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    T.G. Kurtz, Representation and approximation of counting processes, in Advances in Filtering and Optimal Stochastic Control, ed. by W.H. Fleming, L.G. Gorostiza. Lecture Notes in Control and Information Sciences, vol. 42, pp. 177–191 (Springer, Berlin, 1982). doi:10.1007/BFb0004537Google Scholar
  29. 29.
    P.D. Lax, R.D. Richtmyer, Survey of the stability of linear finite difference equations. Commun. Pure Appl. Anal. 9(2), 267–293 (1956). doi:10.1002/cpa.3160090206MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    T. Li, Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6(2), 417–436 (2007). doi:10.1137/06066792XMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    S.P. Meyn, R.L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3), 518–548 (1993)MathSciNetMATHGoogle Scholar
  32. 32.
    B. Øksendal, T. Zhang, The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures. Osaka J. Math. 44(1), 207–230 (2007)MathSciNetMATHGoogle Scholar
  33. 33.
    P.E. Protter, Stochastic Integration and Differential Equations. number 21 in Stochastic Modelling and Applied Probability, 2nd edn. (Springer, Berlin, 2005)Google Scholar
  34. 34.
    M. Rathinam, Moment growth bounds on continuous time Markov processes on non-negative integer lattices. Quart. Appl. Math. 73(2), 347–364 (2015). doi:10.1090/S0033-569X-2015-01372-7MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications. Mathematical and Analytical Techniques with Applications to Engineering (Springer, New York, 2005)Google Scholar
  36. 36.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry, 2nd edn. (Elsevier, Amsterdam, 2004)MATHGoogle Scholar
  37. 37.
    C.A. Yates, M.B. Flegg, The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion. J. R. Soc. Interface 12(106) (2015). doi:10.1098/rsif.2015.0141Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala University05 UppsalaSweden

Personalised recommendations