Model Approximation via Dimension Reduction

In the initial stages of refining a mathematical model of a real-world dynamical system, one is often confronted with many more variables and coupled differential equations than one intuitively feels should be sufficient to describe the system. Yet none of the variables may seem so irrelevant as to be excludable nor so dominant as to explain the overall dynamics. Part of the problem might even be that one has been forced to formulate the problem in some “convenient” but not necessarily “ideal” set of variables. In such a circumstance one wishes to simplify the model by an approximation. In this paper we present a numerical technique called Extended Adiabatic Elimination (so named because it generalizes Direct Adiabatic Elimination) for automatically approximating a dynamical model by an equivalent one involving fewer, more appropriate, variables. Given, a set of coupled ordinary differential equations and a spatial domain of interest, EAE first tests whether the model is approximable and if so, returns the approximate model whose independent variables are called order parameters, together with an indication of the temporal domain of validity of the approximation. The order parameters are composite variables, built from those in the original model, and represent, in a sense, an “ideal” set of variables for the given problem. We explain the theoretical basis underpinning EAE and describe the steps in the procedure with respect to a running example. EAE is both more accurate than DAE and is capable of tackling phase spaces of dimensions beyond those that DAE can handle. Currently, the automated parts of the system can deal with low dimensional phase spaces but, in principle, the algorithm appears to be readily generalisable.