Quasi-equilibrium grid algorithm: Geometric construction for model reduction

Abstract

The method of invariant grid (MIG) is an iterative procedure for model reduction in chemical kinetics which is based on the notion of Slow Invariant Manifold (SIM) [A.N. Gorban, I.V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci. 58 (2003) 4751-4768; E. Chiavazzo, A.N. Gorban, I.V. Karlin, Comparison of invariant manifolds for model reduction in chemical kinetics, Commun. ]. Important role, in that method, is played by the initial grid which, once refined, gives a description of the invariant manifold: the invariant grid. A convenient way to get a first approximation of the SIM is given by the spectral quasi-equilibrium manifold (SQEM) [A.N. Gorban, I.V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci. 58 (2003) 4751-4768; E. Chiavazzo, A.N. Gorban, I.V. Karlin, Comparison of invariant manifolds for model reduction in chemical kinetics, Commun. Comput. Phys. 2(5) (2007) 964-992].In the present paper, a flexible numerical method to construct the discrete analog of a quasi-equilibrium manifold, in any dimension, is presented. That object is named quasi-equilibrium grid (QEG), while the procedure quasi-equilibrium grid algorithm (QEGA). Extensions of the QEM notion are also suggested. The QEGA is a numerical tool which can be used to find a grid-based approximation for the locus of minima of a convex function under some linear constraints. The method is validated by construction of one and two-dimensional grids for a model of hydrogen oxidation reaction.

Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems

“…In this section we consider a small test mechanism, which has already been used for model reduction purposes in [31,36,37]. It consists of six chemical species involved in six elementary reactions constrained by two element mass conservation relations for hydrogen and oxygen: …”

Reaction Kinetics Path Based on Entropy Production Rate and Its Relevance to Low-Dimensional Manifolds

“…For example, Chiavazzo and Karlin [41] devised an algorithm to utilize the so-called quasi-equilibrium manifold as a first approximation to SIM. Regarding this hierarchy of manifolds of lower dimension [18][19][20][21][22][23][26][27][28], the observation that the paths determined by ELNEP trace at least final part of the 1-D and (possibly) 2-D manifolds of true chemical kinetics in the samples described in Section 3 is proved or, at least, theoretically justified to hold in more general cases where the method of intrinsic low-dimensional manifolds (ILDM) [26,39] is applicable.…”

The Role of Thermodynamics in Model Reduction When Using Invariant Grids

“…Quasi-equilibrium manifolds provide reasonably good initial approximations of slow invariant manifolds, and the rationale behind is discussed in Section 2.2. Moreover, comparisons between QEM and SIM can be also found in the literature [16,18].…”

“…For the sake of completeness, below we briefly review the quasi-equilibrium grid algorithm [18] for constructing a grid based approximation of a QEM. Let E be the (d+ q)×n matrix, constructed by adding the a j vectors as q additional rows to the matrix D. Let the steady state of (3.6) be denoted by …”

“…Here, the choice of R is done according to the spectral quasi-equilibrium parametrization suggested in [16,18]. Therefore, R denotes the left eigenvector, of the Jacobi matrix at the equilibrium, which corresponds to the smallest eigenvalue (by absolute value).…”