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Preconditioned Bayesian Regression for Stochastic Chemical Kinetics

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Abstract

We develop a preconditioned Bayesian regression method that enables sparse polynomial chaos representations of noisy outputs for stochastic chemical systems with uncertain reaction rates. The approach is based on the definition of an appropriate multiscale transformation of the state variables coupled with a Bayesian regression formalism. This enables efficient and robust recovery of both the transient dynamics and the corresponding noise levels. Implementation of the present approach is illustrated through applications to a stochastic Michaelis–Menten dynamics and a higher dimensional example involving a genetic positive feedback loop. In all cases, a stochastic simulation algorithm (SSA) is used to compute the system dynamics. Numerical experiments show that Bayesian preconditioning algorithms can simultaneously accommodate large noise levels and large variability with uncertain parameters, and that robust estimates can be obtained with a small number of SSA realizations.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 9th edn. Dover, New York (1972)

    MATH  Google Scholar 

  2. Alexanderian, A., Le Maître, O.P., Najm, H.N., Iskandarani, M., Knio, O.M.: Multiscale stochastic preconditioners in non-intrusive spectral projection. J. Sci. Comput. 50, 306–340 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrieu, C., Moulines, E.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16, 1462–1505 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Atchadé, Y.F., Rosenthal, J.S.: On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11, 815–828 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bennett, M.R., Volfson, D., Tsimring, L., Hasty, J.: Transient dynamics of genetic regulatory networks. Biophys. J. 92(10), 3501–3512 (2007)

    Article  Google Scholar 

  6. Box, G.E.P., Tiao, G.C.: Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MA, USA (1973). [Addison-Wesley Series in Behavioral Science: Quantitative Methods]

  7. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of fourier-hermite functionals. Ann. Math. 48, 385–392 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carlin, B.P., Louis, T.A.: Bayesian Methods for Data Analysis, Texts in Statistical Science Series, 3rd edn. CRC Press, Boca Raton, FL, USA (2009)

    Google Scholar 

  9. Crestaux, T., Le Maitre, O.P., Martinez, J.-M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009). Special Issue on Sensitivity Analysis

  10. Liu, W.E.D., Vanden-Eijnden, E.: Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221(1), 158–180 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ernst, O.G., Mugler, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math Model Numer Anal 46, 317–339 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence [Wiley Series in Probability and Mathematical Statistics] (1986)

  13. Tempone, R., Nobile, F., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ganapathysubramanian, B., Zabaras, N.: Sparse grid collacation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis. Texts in Statistical Science Series, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL, USA (2004)

    Google Scholar 

  16. Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algor. 18, 209–232 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach, 2nd edn. Dover, New York (2002)

  18. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434 (1976)

    Article  MathSciNet  Google Scholar 

  19. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)

    Article  Google Scholar 

  20. Gillespie, D.T.: A rigorous derivation of the chemical master equation. Phys. A: Stat. Mech. Appl. 188(13), 404–425 (1992)

    Article  MathSciNet  Google Scholar 

  21. Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)

    Article  Google Scholar 

  22. Gillespie, D.T., Petzold, L.R.: Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119(16), 8229–8234 (2003)

    Article  Google Scholar 

  23. Haario, H., Saksman, E., Tamminen, J.: An adaptive metropolis algorithm. Bernoulli 7, 223–242 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  25. Keese, A.: Numerical Solution of Systems with Stochastic Uncertainties: A General Purpose Framework for Stochastic Finite Elements. PhD thesis, Tech. Univ. Braunschweigh (2004)

  26. Keese, A., Matthies, H.G.: Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations. Technical report, Institute of Scientific Computing TU Braunschweig Brunswick (2003)

  27. Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics [Scientific Computation]. Springer, New York (2010)

    Google Scholar 

  28. Le Maître, O.P., Mathelin, L., Knio, O.M., Hussaini, M.Y.: Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discret. Continu. Dyn. Syst. 28(1), 199–226 (2010)

    Article  MATH  Google Scholar 

  29. Le Maître, O.P., Najm, H.N., Pebay, P.P., Ghanem, R.G., Knio, O.M.: Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29(2), 864–889 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228(8), 3084–3113 (2009)

    Google Scholar 

  31. Najm, H.N.: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech. 41, 35–52 (2009)

    Article  MathSciNet  Google Scholar 

  32. Najm, H.N., Debusschere, B., Marzouk, Y., Widmer, S., Le Maître, O.P.: Uncertainty quantification in chemical systems. Int. J. Numer. Eng. 80(6), 789–814 (2009)

    Article  MATH  Google Scholar 

  33. Petras, K.: On the smolyak cubature error for analytic functions. Adv. Comput. Math. 12, 71–93 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  34. Petras, K.: Fast calculation in the smolyak algorithm. Numer. Algor. 26, 93–109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. J. Chem. Phys. 119(24), 12784–12794 (2003)

    Article  Google Scholar 

  36. Reagan, M.T., Najm, H.N., Debusschere, B.J., Le Maître, O.P., Knio, O.M., Ghanem, R.G.: Spectral stochastic uncertainty quantification in chemical systems. Combust. Theory Model. 8, 607–632 (2004)

    Article  Google Scholar 

  37. Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: Uncertainty quantification in reacting flow simulations through non-intrusive spectral projection. Combust. Flame 132, 545–555 (2003)

    Article  Google Scholar 

  38. Rizzi, F., Najm, H.N., Debusschere, B.J., Sargsyan, K., Salloum, M., Adalsteinsson, H., Knio, O.M.: Uncertainty quantification in MD simulations. Part I: Forward propagation. Multiscale Model. Simul. 10(4), 1428–1459 (2012)

    Google Scholar 

  39. Rizzi, F., Najm, H.N., Debusschere, B.J., Sargsyan, K., Salloum, M., Adalsteinsson, H., Knio, O.M.: Uncertainty quantification in MD simulations. Part II: Bayesian inference of force-field parameters. Multiscale Model. Simul. 10(4), 1460–1492 (2012)

    Google Scholar 

  40. Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18, 349–367 (2009)

    Article  MathSciNet  Google Scholar 

  41. Salloum, M., Sargsyan, K., Jones, R., Debusschere, B., Najm, H.N., Adalsteinsson, H.: A stochastic multiscale coupling scheme to account for sampling noise in atomistic-to-continuum simulations. Multiscale Model. Simul. 10, 550–584 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  42. Saltelli, A.: Sensitivity analysis for importance assessment. Risk Anal. 22(3), 579–590 (2002)

    Article  Google Scholar 

  43. Sargsyan, K., Debusschere, B., Najm, H.N., Le Maître, O.P.: Spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning. SIAM J. Sci. Comput. 31, 4395–4421 (2010)

    Article  MATH  Google Scholar 

  44. Sargsyan, K., Debusschere, B., Najm, H.N., Marzouk, Y.: Bayesian inference of spectral expansions for predictability assessment in stochastic reaction networks. J. Comput. Theor. Nanosci. 6(10), 2283–2297 (2009)

    Article  Google Scholar 

  45. Sargsyan, K., Safta, C., Debusschere, B., Najm, H.: Multiparameter spectral representation of noise-induced competence in bacillus subtilis. IEEE/ACM Trans. Comput. Biol. Bioinform. (2012). doi:10.1109/TCBB.2012.107

  46. Schlögl, F.: On thermodynamics near a steady state. Zeitschrift für Physik A Hadrons and Nuclei 248(5), 446–458 (1971)

    Google Scholar 

  47. Sheen, D.A., Wang, H.: Combustion kinetic modeling using multispecies time histories in shock-tube oxidation of heptane. Combust. Flame 158, 645–656 (2011)

    Article  Google Scholar 

  48. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)

    Google Scholar 

  49. Sobol, I.M.: Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Math. Comput. Simul. 55(1–3), 271–280 (2001). The Second IMACS Seminar on Monte Carlo Methods

  50. Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)

    Article  Google Scholar 

  51. Villegas, M., Augustin, F., Gilg, A., Hmaidi, A., Wever, U.: Application of the polynomial chaos expansion to the simulation of chemical reactors with uncertainties. Math. Comput. Simul. 82, 805–817 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  52. Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    Article  MathSciNet  Google Scholar 

  53. Wilkinson, D.J.: Stochastic Modelling for Systems Biology [Chapman & Hall/CRC Mathematical and Computational Biology Series]. Chapman & Hall/CRC, Boca Raton, FL, USA (2006)

    Google Scholar 

  54. Xiu, D.B., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the US Department of Energy (DOE) under Award Numbers DE-SC0001980 and DE-SC0008789. The work of OLM is partially supported by the French Agence Nationale pour la Recherche (Project ANR-2010-Blan-0904) and the GNR MoMaS funded by Andra, Brgm, Cea, Edf, and Irsn. Finally, would like to thank the reviewer for helpful comments on improving this manuscript.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Alen Alexanderian

  2. Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA

    Francesco Rizzi & Omar M. Knio

  3. Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Muruhan Rathinam

  4. LIMSI-CNRS, BP 133, Bt 508, 91403 , Orsay Cedex, France

    Olivier P. Le Maître

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  1. Alen Alexanderian
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  4. Olivier P. Le Maître
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Correspondence to Omar M. Knio.

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Alexanderian, A., Rizzi, F., Rathinam, M. et al. Preconditioned Bayesian Regression for Stochastic Chemical Kinetics. J Sci Comput 58, 592–626 (2014). https://doi.org/10.1007/s10915-013-9745-5

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