A general analysis of approximate lumping in chemical kinetics

Abstract

A general analysis of approximate lumping is presented. This analysis can be applied to any reaction system with n species described by d y/d t = f(y), where y is an n-dimensional vector in a desired region Ω, and f(y) is an arbitrary n-dimensional function vector. Here we consider lumping by means of a rectangular constant matrix M (i.e. ŷ = My, where M is a row-full rank matrix and ŷ has dimension n̂ not larger than n). The observer theory initiated by Luenberger is formally employed to obtain the kinetic equations and discuss the properties of the approximately lumped system. The approximately lumped kinetic equations have the same form d ŷ/d t = M f/ M y) as that for exactly lumped ones, but depend on the choice of the generalized inverse M of M. {1,2,3,4}-inverse is a good choice of the generalized inverse of M. The equations to determine the approximate lumping matrices M are presented. These equations can be solved by iteration. An approach for choosing suitable initial iteration values of the equations is illustrated by examples.