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Applications of Magnus expansions and pseudospectra to Markov processes

Published online by Cambridge University Press:  17 April 2018

A. ISERLES  and
S. MACNAMARA
A. ISERLES
Affiliation:
Department of Applied Mathematics and Mathematical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK email: [email protected]
S. MACNAMARA
Affiliation:
Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, Australia email: [email protected]
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Abstract

New directions in Markov processes and research on master equations are showcased by example. The utility of Magnus expansions for handling time-varying rates is demonstrated. The useful notion in applied mathematics often turns out to be the pseudospectra and not simply the eigenvalues. We highlight that general principle with our own examples of Markov processes where exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods. As a motivating application, isomerisation provides a running example and an illustration of our approaches to chemical kinetics. We also present a brief example of a totally asymmetric exclusion process.

Keywords

60J2860H3565F40
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 400 - 425
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†This research and Shev MacNamara have been partially supported by a David G. Crighton Fellowship to DAMTP, Cambridge.

References

[1] Al-Mohy, A. H. & Higham, N. J. (2009) A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl. 31 (3), 970–989.Google Scholar
[2] Al-Mohy, A. H. & Higham, N. J. (2011) Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comp. 33, 488–511.Google Scholar
[3] Anderson, D. (2012) An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM J. Numer. Anal. 50, 2237–2258.Google Scholar
[4] Anderson, D. & Kurtz, T. (2011) Continuous time Markov chain models for chemical reaction networks. In: Koeppl, H., Setti, G., di Bernardo, M., Densmore, D. (editors), Design and Analysis of Biomolecular Circuits, New York: Springer.Google Scholar
[5] Barker, J. R., Nguyen, T. L., Stanton, J. F., Aieta, M. C. C., Gabas, F., Kumar, T. J. D., Li, C. G. L., Lohr, L. L., Maranzana, A., Ortiz, N. F., Preses, J. M. & Stimac, P. J. (2016) Multiwell-2016 Software Suite, Technical Report, University of Michigan, Ann Arbor, Michigan, USA.Google Scholar
[6] Bolley, C. & Crouzeix, M. (1978) Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques. RAIRO Anal. Numér. 12, 237–245.Google Scholar
[7] Celledoni, E. & Iserles, A. (2001) Methods for the approximation of the matrix exponential in a Lie-algebraic setting. IMA J. Numer. Anal. 21, 463–488.Google Scholar
[8] Corwin, I. (2014) Macdonald processes, quantum integrable systems and the Kardar–Parisi–Zhang universality class. In: Proceedings of the International Congress of Mathematicians.Google Scholar
[9] Corwin, I. (2016) Kardar–Parisi–Zhang Universality. Notices of the American Mathematical Society, March 2016 Not. AMS 63.Google Scholar
[10] Drawert, B., Trogdon, M., Toor, S., Petzold, L. & Hellander, A. (2016) Molns: A cloud platform for interactive, reproducible, and scalable spatial stochastic computational experiments in systems biology using pyurdme. SIAM J. Sci. Comput. 38, C179–C202.Google Scholar
[11] Edelman, A. & Kostlan, E. (1994) The Road from Kac's Matrix to Kac's Random Polynomials, Technical Report, University of California, Berkeley.Google Scholar
[12] Edelman, A. & Rao, N. R. (2005) Random matrix theory. Acta Numer. 14, 233–297.Google Scholar
[13] Evans, S. N., Sturmfels, B. & Uhler, C. (2010) Commuting birth-and-death processes. Ann. Appl. Probab. 20, 238–266.Google Scholar
[14] Giles, M. & Glasserman, P. (2006) Smoking adjoints: Fast Monte Carlo Greeks. Risk 19 (1), 88–92.Google Scholar
[15] Gillespie, D. T. (2002) The chemical Langevin and Fokker–Planck equations for the reversible isomerization reaction. J. Phys. Chem. A 106, 5063–5071.Google Scholar
[16] Gorenflo, R., Kilbas, A., Mainardi, F. & Rogosin, S. (2014) Mittag-Leffler Functions, Related Topics and Applications, New York: Springer.Google Scholar
[17] Gunawardena, J. (2012) A linear framework for time-scale separation in nonlinear biochemical systems. PLoS One 7, e36321.Google Scholar
[18] Gunawardena, J. (2014) Time-scale separation: Michaelis and Menten's old idea, still bearing fruit. FEBS J. 281, 473–488.Google Scholar
[19] Hairer, M. (2014) Singular stochastic PDEs. In: Proceedings of the International Congress of Mathematicians.Google Scholar
[20] Hellander, A., Klosa, J., Lötstedt, P. & MacNamara, S. (2017) Robustness analysis of spatiotemporal models in the presence of extrinsic fluctuations. SIAM J. Appl. Math. 77 (4), 1157–1183.Google Scholar
[21] Higham, D. J. (2008) Modeling and simulating chemical reactions. SIAM Rev. 50, 347–368.Google Scholar
[22] Hilfinger, A. & Paulsson, J. (2011) Separating intrinsic from extrinsic fluctuations in dynamic biological systems. Proc. Acad. Natl. Sci. 109, 12167–72.Google Scholar
[23] Hilgers, P. V. & Langville, A. N. (2006) The five greatest applications of Markov chains. In: Proceedings of the Markov Anniversary Meeting, Boston Press, Boston, MA.Google Scholar
[24] Hochbruck, M. & Lubich, C. (2003) On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 41, 945–963.Google Scholar
[25] Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. & Zanna, A. (2000) Lie-group methods. Acta Numer. 9, 215–365.Google Scholar
[26] Jahnke, T. & Huisinga, W. (2007) Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54, 1–26. cited By 97.Google Scholar
[27] Kac, M. (1957) Probability and Related Topics in Physical Sciences, Summer Seminar in Applied Mathematics, American Mathematical Society, Boulder, Colorado.Google Scholar
[28] Kormann, K. & MacNamara, S. (2016) Error control for exponential integration of the master equation, Technical Report.Google Scholar
[29] Kurtz, T. (1980) Representations of Markov processes as multiparameter time changes. Ann. Probab. 8, 682–715.Google Scholar
[30] Leite, S. C. & Williams, R. J. (2016) A constrained Langevin approximation for chemical reaction networks. Kolmogorov Lecture, 9th World Congress In Probability and Statistics, Toronto, http://www.math.ucsd.edu/williams/biochem/biochem.pdf, (2016).Google Scholar
[31] Macnamara, S. (2015) Cauchy integrals for computational solutions of master equations. ANZIAM J. 56, 32–51.Google Scholar
[32] MacNamara, S., Burrage, K. & Sidje, R. (2008) Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. Sim. 6, 1146–1168.Google Scholar
[33] MacNamara, S., Henry, B. I. & McLean, W. (2016) Fractional Euler limits and their applications. SIAM J. Appl. Math.Google Scholar
[34] MacNamara, S. & Strang, G. (2015) Master equations in ‘Essays on New Directions in Numerical Computation’, http://tobydriscoll.net/newdirections2015/, http://tobydriscoll.net/newdirections2015/LNT60Essays.pdf.Google Scholar
[35] Magnus, W. (1954) On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7, 649–673.Google Scholar
[36] Moler, C. & Loan, C. V. (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49.Google Scholar
[37] Munthe-Kaas, H. Z., Quispel, G. R. W. & Zanna, A. (2001) Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1, 297–324.Google Scholar
[38] Pavliotis, G. A. & Stuart, A. (2008) Multiscale Methods: Averaging and Homogenization, New York: Springer.Google Scholar
[39] Reddy, S. C. & Trefethen, L. N. (1994) Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math.Google Scholar
[40] Strang, G. & MacNamara, S. (2014) Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev. 56, 525–546.Google Scholar
[41] Timm, C. (2009) Random transition-rate matrices for the master equation. Phys. Rev. E 80, 021140, New Jersey.Google Scholar
[42] Trefethen, L. N. & Chapman, S. J. (2004) Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57, 1233–1264.Google Scholar
[43] Trefethen, L. N. & Embree, M. (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press.Google Scholar
[44] Weber, M. F. & Frey, E. (2016) Master equations and the theory of stochastic path integrals, Rep Prog Phys. 2017 Apr; 80 (4):046601. doi: 10.1088/1361-6633/aa5ae2.Google Scholar
[45] Wei, J. & Norman, E. (1964) On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc. 15, 327–334.Google Scholar
[46] Wright, T. G. (2002) Eigtool, http://www.comlab.ox.ac.uk/pseudospectra/eigtool/,2002.Google Scholar