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Reduction of Linear Systems of ODEs with Optimal Replacement Variables

Published online by Cambridge University Press:  20 August 2015

Alex Solomonoff  and
Wai Sun Don
Alex Solomonoff*
Affiliation:
Camberville Research Institute, Somerville, MA, USA
Wai Sun Don*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
Corresponding author.Email:[email protected]
Email:[email protected]
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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.

Keywords

65M0665M3065M70System reductionoptimal replacement variablesresolved variablesoptimal prediction
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 756 - 779
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Chorin, A.J. and Stinis, P., Problem reduction, renormalization, and memory, Commun. Appl. Math. Comput. Sci., 1(1) (2005), 1–27.Google Scholar
[2]Chorin, A. J. and Hald, O. H., Stochastic Tools in Mathematics and Science, Springer, 2006.Google Scholar
[3]Chertock, A., Gottlieb, D. and Solomonoff, A., Modified optimal prediction and its application to a particle-method problem, J. Sci. Comput., 37 (2008), 189–201.Google Scholar
[4]Givon, D., Kupferman, R. and Stuart, A., Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity., 17(6) (2004), R55–R127, MR 2097022.Google Scholar
[5]Traub, J. F., Wasilkowski, G. W. and Wozniakowski, H., Information-Based Complexity, Academic Press, 1988.Google Scholar
[6]Yang, C., Meza, J. C. and Wang, L.-W., A trust region direct constrained minimization algorithm for the Kohn-Sham equation, SIAM J. Sci. Comput., 29(5) (2007), 1854–1875.Google Scholar
[7]Edelman, A., Arias, T. A. and Smith, S. T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix. Anal. Appl., 20(2) (1998), 303–353.Google Scholar
[8]Searle, S. R., Linear Models, Wiley and Sons, 1997.Google Scholar
[9]Fortmann, T., A matrix inversion identity, IEEE Trans. Auto. Control., 15(5) (1970), 599–599.Google Scholar
[10]Eaton, John W., GNU Octave Manual, Network Theory Limited, 2002, isbn=0-9541617-2-6, also www.octave.org.Google Scholar
[11]Cover, T. M. and Thomas, J. A., Determinant inequalities via information theory, SIAM J. Matrix. Anal. Appl. 9(3) (1988), 384–392.Google Scholar