TY - JOUR
T1 - Reduction of Linear Systems of ODEs with Optimal Replacement Variables
AU - Alex Solomonoff & Wai Sun Don
JO - Communications in Computational Physics
VL - 3
SP - 756
EP - 779
PY - 2011
DA - 2011/03
SN - 9
DO - http://doi.org/10.4208/cicp.271109.150710s
UR - https://global-sci.org/intro/article_detail/cicp/7520.html
KW - 
AB - <p style="text-align: justify;">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.</p>