Volume 9, Issue 3
Reduction of Linear Systems of ODEs with Optimal Replacement Variables

Alex Solomonoff & Wai Sun Don

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

Published online: 2011-03

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  • Abstract

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.

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@Article{CiCP-9-756, author = {}, title = {Reduction of Linear Systems of ODEs with Optimal Replacement Variables}, journal = {Communications in Computational Physics}, year = {2011}, volume = {9}, number = {3}, pages = {756--779}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.271109.150710s}, url = {http://global-sci.org/intro/article_detail/cicp/7520.html} }
TY - JOUR T1 - Reduction of Linear Systems of ODEs with Optimal Replacement Variables 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 -

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.

Alex Solomonoff & Wai Sun Don. (2020). Reduction of Linear Systems of ODEs with Optimal Replacement Variables. Communications in Computational Physics. 9 (3). 756-779. doi:10.4208/cicp.271109.150710s
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