A continuous genetic algorithm for the numerical calibration in scalar conservation laws


  • Stefan Berres Departamento de Ciencias Matemáticas, Facultad de Ingeniería, Universidad Católica de Temuco, Rudecindo Ortega 02950, Temuco-Chile.
  • Aníbal Coronel Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío Bío, Avda. Andrés Bello 720, Chillán-Chile.
  • Richard Lagos Departamento de Matemática y Física, Facultad de Ciencias, Universidad de Magallanes, Av. Bulnes 01855, Punta Arenas-Chile.




Finite volume, genetic algorithm, flux identification, conservation law


Our work deals with the flux identication problem for scalar conservation laws. The problem is formulated as an optimization problem, where the objective function compares the solution of the
direct problem with observed proles at a fixed time. A finite volume scheme solves the direct problem and a continuous genetic algorithm solves the inverse problem. The numerical method is tested with synthetic experimental data. Simulation parameters are recovered approximately. The tested heuristic optimization technique turns out to be more robust than classical optimization techniques.


S. Berres, A. Coronel, R. Lagos and M. Sepúlveda. Performance of a real coded genetic algorithm the calibration of scalar conservation laws. Anziam J., 58:51–77, 2016.

C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer-Verlag, Berlin, third edition, 2010.

J. De Clerq, I. Nopens, J. Defrancq and Pa. Vanrolleghem. Extending and calibrating a mechanistic hindered and compression settling model for activated sludge using indepth batch experiments. Water Research, 42(3):781–791, 2008.

R. Eymard, T. Gallouet and R. Herbin. Finite Volume Methods. In Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, pages 713–1020. North-Holland, Amsterdam, 2000.

L.J. Fogel, A.J. Owens and M.J.Walsh. Artificial Intelligence Through Simulated Evolution. Wiley, Chichester, WS, UK, 1966.

R. L. Haupt and S. E. Haupt. Practical Genetic Algorithms. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, second edition, 2004.

H. Holden, F.S. Priuli, and N.H. Risebro. On an inverse problem for scalar conservation laws. Inverse Problems,30(3):035015, 2014.

J.H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI, USA, 1975.

J.R. Koza. Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA, USA, 1992.

J.C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions, SIAM Journal of Optimization, 9(1):112–147, 1998.

R. Lagos. Estudio analítico y numérico de un problema inverso originado en la extracción secundaria de petróleo. Tesis de Magíster, Universidad del Bío-Bío, 2015.

R.J. LeVeque. Numerical methods for conservation laws. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, second edition, 1992.

M. J. Lighthill and G. B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A., 229:317–345, 1955.

H. Liu and T. Pan. Interaction of elementary waves for scalar conservation laws on a bounded domain. Math. Methods Appl. Sci., 26(7):619–632, 2003.

I. Rechenberg. Evolutionstrategie: Optimierung Technischer Systeme nach Prinzipien des Biologischen Evolution. Frommann-Holzboog Verlag, Stuttgart, 1973.

P. Rocca, M. Benedetti, M. Donelli, D. Franceschini and A. Massa. Evolutionary optimization as applied to inverse scattering problems. Inverse Problems, 25(12):123003, 2009.

S.N. Sivanandam and S. N. Deepa. Introduction to Genetic Algorithms. Springer, Berlin, 2008.

E.F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics : A practical introduction. Springer-Verlag, Berlin, third edition, 2009.



How to Cite

Berres, S., Coronel, A., & Lagos, R. (2017). A continuous genetic algorithm for the numerical calibration in scalar conservation laws. Selecciones Matemáticas, 4(01), 16-24. https://doi.org/10.17268/sel.mat.2017.01.02