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Weak error analysis of tau-leaping methods for multi-scale stochastic chemical kinetic systems
 

Summary: Weak error analysis of tau-leaping methods for multi-scale
stochastic chemical kinetic systems
David F. Anderson1
and Masanori Koyama2
June 20, 2011
Abstract
A chemical reaction network is a chemical system involving multiple reactions and chemical
species. The simplest stochastic models of such networks treat the system as a continuous
time Markov chain with the state being the number of molecules of each species and with
reactions modeled as possible transitions of the chain. For such models, there is typically a wide
variation in temporal and other quantitative scales. In this multi-scale setting it is typically an
extremely difficult task to perform approximations, such as Langevin approximations or law of
large number type arguments, to simplify a system. Therefore, numerical methods oftentimes
are the only reasonable means by which such models can be understood in real time.
In this paper we provide a general framework for understanding the weak error of numerical
approximation techniques in the multi-scale setting. We quantify how the error of three different
methods depends upon both the natural scalings within a given system, and with the step-size
of the numerical method. Further, we introduce a new algorithm in this setting, the weak
trapezoidal algorithm, which was developed originally as an approximate method for diffusion
processes, and prove that the leading order of the error process scales with the square of the

  

Source: Anderson, David F. - Department of Mathematics, University of Wisconsin at Madison

 

Collections: Mathematics