Commun. Comput. Phys., 9 (2011), pp. 756-779.

Reduction of Linear Systems of ODEs with Optimal Replacement Variables

Alex Solomonoff 1*, Wai Sun Don 2

1 Camberville Research Institute, Somerville, MA, USA.
2 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.

Received 27 November 2009; Accepted (in revised version) 15 July 2010
Available online 17 September 2010


In this exploratory study, we present a new method of approximating a large system of ODEs by one with fewer equations, while attempting to preserve the essential dynamics of a reduced set of variables of interest. The method has the following key elements: (i) put a (simple, ad-hoc) probability distribution on the phase space of the ODE; (ii) assert that a small set of replacement variables are to be unknown linear combinations of the not-of-interest variables, and let the variables of the reduced system consist of the variables-of-interest together with the replacement variables; (iii) find the linear combinations that minimize the difference between the dynamics of the original system and the reduced system. We describe this approach in detail for linear systems of ODEs. Numerical techniques and issues for carrying out the required minimization are presented. Examples of systems of linear ODEs and variable-coefficient linear PDEs are used to demonstrate the method. We show that the resulting approximate reduced system of ODEs gives good approximations to the original system. Finally, some directions for further work are outlined.

AMS subject classifications: 65M06, 65M30, 65M70

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Key words: System reduction, optimal replacement variables, resolved variables, optimal prediction.

*Corresponding author.
Email: (A. Solomonoff), (W.-S. Don)

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