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System Reduction Using an LQR-Inspired Version of Optimal Replacement Variables

Published online by Cambridge University Press:  20 August 2015

Alex Solomonoff
Alex Solomonoff*
Affiliation:
Camberville Research Institute, Somerville, MA, USA Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
*Corresponding author.Email address:alex.solomonoff@yahoo.com
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Abstract

Optimal Replacement Variables (ORV) is a method for 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. An earlier version of ORV [1] had some issues, including limited accuracy and in some rare cases, instability. Here we present a new version of ORV, inspired by the linear quadratic regulator problem of control theory, which provides better accuracy, a guarantee of stability and is in some ways easier to use.

Keywords

65M0665M3065M70System reductionoptimal replacement variablesresolved variablesoptimal prediction
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1520 - 1540
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Solomonoff, A. and Don, Wai Sun, Reduction of linear systems of ODES with optimal replacement variables, Commun. Comput. Phys., 9 (2011), 756–779.Google Scholar
[2]Bartels, R. H. and Stewart, G. W., Solution of the matrix equation AX + XB = C, Commun. ACM, 15(9) (1972), 820–826.CrossRefGoogle Scholar
[3]G, Steven. Johnson, The NLopt nonlinear-optimization package, ab-initio. mit.edu/nlopt.Google Scholar
[4]Eaton, John W., GNU Octave Manual, Network Theory Limited, 2002, ISBN=0-9541617-2-6, also www.octave.org.Google Scholar
[5]Chorin, A., Hald, O. H. and Kupferman, R., Optimal prediction with memory, Phys. D, 166 (2002), 239–257.Google Scholar
[6]Chorin, A. and Stinis, P., Problem reduction, renormalization and memory, Commun. Appl. Math. Comput. Sci., 1(1) (2005), 1–27.Google Scholar
[7]Chorin, A. J. and Hald, O. H., Stochastic Tools in Mathematics and Science, Springer, 2006Google Scholar
[8]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
[9]Givon, D., Kupferman, R. and Stuart, A., Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity, 17(6) (2004), R55–R127, MR 2097022.CrossRefGoogle Scholar
[10]Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, Wiley-Interscience, 1972.Google Scholar
[11]Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis, Cambridge University Press, 1994.Google Scholar
[12] Wikipedia contributors, Trace (linear algebra), Wikipedia, the free encyclopedia.Google Scholar
[13]Sirovich, L., Analysis of turbulent flows by means of the empirical eigenfunctions, Fluid Dyn. Res., 8 (1991), 85–100.Google Scholar
[14]Brandts, J., A comparison of subspace methods for sylvester equations, preprint No. 1183, Universiteit Utrecht Mathematics Institute, 2001.Google Scholar
[15]Benner, P., Quintana-Orti, E. S., Quintana-Orti, G., Efficient numerical algorithms for balanced stochastic truncation, 2001, see also http: //matwbn. icm.edu.pl/ksiazki/amc/amc11/amc1 156.pdf.Google Scholar
[16]Antoulas, A., Sorensen, D., Gallivan, K. A., Dooren, P. Van, Grama, A., Hoffman, C. and Sameh, A., Model Reduction of Large-Scale Dynamical Systems, Computational Science-ICCS 2004,4th International Conference, Krakow, Poland, June 6–9, 2004, Proceedings, Part III, Bubak, M., van Albada, G. Dick, Sloot, P. M. A. and Dongarra, J., eds., Lecture Notes in Computer Science, 3038 (2004), 740–747.Google Scholar