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Tensor approximation of stationary distributions of chemical reaction networks

by V. Kazeev and Ch. Schwab

(Report number 2013-18)

Abstract
We prove that the stationary distribution of a system of reacting species with a weakly- reversible reaction network of zero deficiency in the sense of Feinberg admits tensor- structured approximation of complexity which scales linearly with respect to the number of species and logarithmically in the maximum copy numbers as well as in the desired accuracy. Our results cover the classical mass-action and also Michaelis-Menten kinetics which correspond to two widely used classes of propensity functions, and also to arbitrary combinations of those. New rank bounds for tensor-structured approximations of the PDF of a truncated one-dimensional Poisson distribution are an auxiliary result of the present paper, which might be of independent interest. The present work complements recent results obtained by the authors jointly with M. Khammash and M. Nip on the tensor-structured numerical simulation of the evolution of system states distributions, driven by the Kolmogorov forward equation of the system, known also as the chemical master equation, or CME for short. For the two kinetics mentioned above we also analyze the low-rank tensor structure of the CME operator.

Keywords: chemical master equation; stochastic models; low rank; tensor approximation; tensor train; quantized tensor train; multilinear algebra; mass-action kinetics; Michaelis–Menten kinetics; stationary distribution; deficiency zero; chemical reaction network; Poisson distribution

@Techreport{KS13_514,
  author = {V. Kazeev and Ch. Schwab},
  title = {Tensor approximation of stationary distributions of chemical reaction networks},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-18.pdf },
  year = {2013}
}

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