Reduction of Linear Systems of ODEs with Optimal Replacement Variables (Communications in Computational Physics, 2011)
In this exploratory study, we present a new method of approximating a largesystem of ODEs by one with fewer equations, while attempting to preserve the essentialdynamics of a reduced set of variables of interest. The method has the followingkey elements: (i) put a (simple, ad-hoc) probability distribution on the phase spaceof the ODE; (ii) assert that a small set of replacement variables are to be unknown linearcombinations of the not-of-interest variables, and let the variables of the reducedsystem consist of the variables-of-interest together with the replacement variables; (iii)find the linear combinations that minimize the difference between the dynamics of theoriginal system and the reduced system. We describe this approach in detail for linearsystems of ODEs. Numerical techniques and issues for carrying out the required minimizationare presented. Examples of systems of linear ODEs and variable-coefficientlinear PDEs are used to demonstrate the method. We show that the resulting approximatereduced system of ODEs gives good approximations to the original system. Finally,some directions for further work are outlined.