Abstract
This research work presents a new evolutionary optimization algorithm, Evo-Runge-Kutta in theoretical mathematics with applications in scientific computing. We illustrate the application of Evo-Runge-Kutta, a two-phase optimization algorithm, to a problem of pure algebra, the study of the parameterization of an algebraic variety, an open problem in algebra. Results show the design and optimization of particular algebraic varieties, the Runge-Kutta methods of order q. The mapping between algebraic geometry and evolutionary optimization is direct, and we expect that many open problems in pure algebra will be modelled as constrained global optimization problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kaw, A., Kalu, E.E.: Numerical Methods with Applications. Autarkaw (2010)
Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equation I: Nonstiff Problems, 3rd edn. Springer (January 2010)
Goldberg, D.: Design of Innovation. Kluwar (2002)
Butcher, J.C.: Numerical methods for ordinary differential equations. John Wiley and Sons (2008)
Famelis, T., Papakostas, S.N., Tsitouras, C.: Symbolic derivation of runge kutta order conditions. J. Symbolic Comput. 37, 311–327 (2004)
Fulton, W.: Algebraic Curves. Addison Wesley Publishing Company (December 1974)
Kollar, J.: Rational Curves on Algebraic Varieties. Springer (1996)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer (2007)
D’Andrea, C., Sombra, M.: The newton polygon of a rational plane curve. arXiv:0710.1103
Castro, D., Giusti, M., Heintz, J., Matera, G., Pardo, L.M.: The hardness of polynomial equation solving. Found. Comput. Math. 3(4), 347–420 (2003)
Giusti, M., Hägele, K., Lecerf, G., Marchand, J., Salvy, B.: The projective Noether Maple package: computing the dimension of a projective variety. J. Symbolic Comput. 30(3), 291–307 (2000)
Giusti, M., Lecerf, G., Salvy, B.: A Gröbner free alternative for polynomial system solving. J. Complexity 17(1), 154–211 (2001)
Greene, R.E., Yau, S.T. (eds.): Open Problems in Geometry. Proceedings of Symposia in Pure Mathematics, vol. 54 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Martino, I., Nicosia, G. (2012). Global Optimization for Algebraic Geometry – Computing Runge–Kutta Methods. In: Hamadi, Y., Schoenauer, M. (eds) Learning and Intelligent Optimization. LION 2012. Lecture Notes in Computer Science, vol 7219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34413-8_43
Download citation
DOI: https://doi.org/10.1007/978-3-642-34413-8_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34412-1
Online ISBN: 978-3-642-34413-8
eBook Packages: Computer ScienceComputer Science (R0)