Abstract
The problem of computing the transient probability distribution of countably infinite multidimensional continuous-time Markov chains (CTMCs) arising in systems of stochastic chemical kinetics is addressed by a software tool. Starting from an initial probability distribution, time evolution of the probability distribution associated with the CTMC is described by a system of linear first-order ordinary differential equations, known as the chemical master equation (CME). The solver for the CME uses the time stepping implicit backward differentiation formulae (BDF). Solution vectors in BDF can be stored compactly during transient analysis in one of the Hierarchical Tucker Decomposition, Quantized Tensor Train, or Transposed Quantized Tensor Train formats.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Buchholz, P., Dayar, T., Kriege, J., Orhan, M.C.: On compact solution vectors in Kronecker-based Markovian analysis. Perform. Eval. 115, 132–149 (2017)
CompactTransientSolver software (2017). http://www.cs.bilkent.edu.tr/~tugrul/software.html
Dayar, T.: Analyzing Markov Chains using Kronecker Products: Theory and Applications. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-4190-8
Goutsias, J., Jenkinson, G.: Markovian dynamics on complex reaction networks. Phys. Rep. 529(2), 199–264 (2013)
Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28027-6
Hegland, M., Burden, C., Santoso, L., MacNamara, S., Booth, H.: A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math. 205(2), 708–724 (2007)
Kazeev, V., Khammash, M., Nip, M., Schwab, C.: Direct solution of the chemical master equation using quantized tensor trains. PLoS Comput. Biol. 10(3), e1003359 (2014)
Khoromskij, B.N.: \(O(d \text{ log }N)\)-Quantics approximation of \(N-d\) tensors in high-dimensional numerical modeling. Constructive Approximation 34(2), 257–280 (2011)
Kressner, D., Tobler, C.: htucker – A Matlab toolbox for tensors in hierarchical Tucker format. Technical Report 2012–02, Mathematics Institute of Computational Science and Engineering, Lausanne (2012)
Orhan, M.C.: On the Numerical Analysis of Infinite Multi-dimensional Markov Chains. PhD Thesis, Department of Computer Engineering, Bilkent University, Ankara (2017)
Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)
Acknowledgement
Part of this work is supported by the Alexander von Humboldt Foundation through the Research Group Linkage Programme. The research of M. Can Orhan is carried out during his PhD studies at Bilkent University and supported by The Scientific and Technological Research Council of Turkey under grant 2211-A. We thank the referees whose comments led to an improved manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Dayar, T., Orhan, M.C. (2018). A Software Tool for the Compact Solution of the Chemical Master Equation. In: German, R., Hielscher, KS., Krieger, U. (eds) Measurement, Modelling and Evaluation of Computing Systems. MMB 2018. Lecture Notes in Computer Science(), vol 10740. Springer, Cham. https://doi.org/10.1007/978-3-319-74947-1_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-74947-1_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74946-4
Online ISBN: 978-3-319-74947-1
eBook Packages: Computer ScienceComputer Science (R0)