Invariant Grids: Method of Complexity Reduction in Reaction Networks

Abstract

Complexity in the description of big chemical reaction networks has both structural (number of species and reactions) and temporal (very different reaction rates) aspects. A consistent way to make model reduction is to construct the invariant manifold which describes the asymptotic system behaviour. In this paper we present a discrete analog of this object: an invariant grid. The invariant grid is introduced independently from the invariant manifold notion and can serve to represent the dynamic system behaviour as well as to approximate the invariant manifold after refinement. The method is designed for pure dissipative systems and widely uses their thermodynamic properties but allows also generalizations for some classes of open systems. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.

Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

“…For the second derivative of ϕ, that appears in (12), the application of the chain rule yields (with •, • 2 being the Euclidian scalar product)…”

“…Some iterative methods came into application that are based on an evaluation of functional equations suitably describing the central characteristics of a slow attracting manifold, for example invariance and stability. Examples are Fraser's algorithm [8][9][10] and the method of invariant grids [11,12,3]. Other widely known and applied methods are e.g.…”

“…In contrast to ILDM and CSP some iterative methods came into application that are not based on explicit computation of a time scale separation, but rather an evaluation of functional equations suitably describing the central characteristics of a slow attracting manifold, for example invariance and stability. Examples are Fraser's algorithm [8][9][10] and the method of invariant grids [3,11,12]. Other widely known and successful methods are the constrained runs algorithm [13,14], the rate-controlled constrained equilibrium (RCCE) method [15], the invariant constrained equilibrium edge preimage curve (ICE-PIC) method [16,17], and flamelet-generated manifolds [18,19].…”

“…Bringing the last two formulae together with (12) and t = u(s) we arrive at the formula for the curvature of c(t):…”

A Variational Principle for Computing Slow Invariant Manifolds in Dissipative Dynamical Systems

“…Among those methods that became popular in applications are the intrinsic low dimensional manifold (ILDM) method [21] and recent extensions of its main ideas, e.g. the global quasi-linearization (GQL) [4], computational singular perturbation (CSP) [15,16], Fraser's algorithm [6,9,23], the method of invariant grids [5,11,12], the constrained runs algorithm [10,33], rate-controlled constrained equilibrium (RCCE) [14], the invariant constrained equilibrium edge preimage curve (ICE-PIC) method [27,28], flamelet-generated manifolds [7,30], and finite time Lyapunov exponents [22]. For a comprehensive overview see e.g.…”