Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2009, Article ID 510934, 14 pages
http://dx.doi.org/10.1155/2009/510934
Research Article

Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

1College of Basic Sciences, Huazhong Agricultural University, Wuhan 430070, China
2College of Engineering and Technology, Huazhong Agricultural University, Wuhan 430070, China

Received 14 December 2008; Revised 13 April 2009; Accepted 30 April 2009

Academic Editor: Slimane Adjerid

Copyright © 2009 Xiaoyan Deng and Qingxi Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, NY, USA, 1997.
  2. B. Malengier and R. van Keer, “Parameter estimation in convection dominated nonlinear convection-diffusion problems by the relaxation method and the adjoint equation,” Journal of Computational and Applied Mathematics, vol. 215, no. 2, pp. 477–483, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. F. Barnsley, Fractal Everywhere, Academic Press, New York, NY, USA, 1988.
  4. M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, no. 4, pp. 303–329, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Deng, H. Li, and X. Chen, “The symbol series expression and Hölder exponent estimates of fractal interpolation function,” Journal of Computational Analysis and Applications, vol. 11, no. 3, pp. 507–523, 2009. View at Google Scholar
  6. Y. Fisher, Fractal Image Compression, Theory and Application, Springer, New York, NY, USA, 1995. View at MathSciNet
  7. A. E. Jacquin, “Image coding based on a fractal theory of iterated contractive image transformations,” IEEE Transactions of Image Processing, vol. 1, no. 1, pp. 18–30, 1992. View at Google Scholar
  8. H. E. Kunze and E. R. Vrscay, “Solving inverse problems for ordinary differential equations using the Picard contraction mapping,” Inverse Problems, vol. 15, no. 3, pp. 745–770, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. E. Kunze, J. E. Hicken, and E. R. Vrscay, “Inverse problems for ODEs using contraction maps and suboptimality of the ‘collage method’,” Inverse Problems, vol. 20, no. 3, pp. 977–991, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. Kunze and K. Heidler, “The collage coding method and its application to an inverse problem for the Lorenz system,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 124–129, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Kunze and S. Vasiliadis, “Using the collage method to solve ODEs inverse problems with multiple data sets,” Nonlinear Analysis: Theory, Methods & Applications. In press. View at Publisher · View at Google Scholar
  12. H. Kunze, D. La Torre, and E. R. Vrscay, “A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems,” Nonlinear Analysis: Theory, Methods & Applications. In press. View at Publisher · View at Google Scholar
  13. X. Deng, B. Wang, and G. Long, “The Picard contraction mapping method for the parameter inversion of reaction-diffusion systems,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2347–2355, 2008. View at Google Scholar · View at MathSciNet
  14. M. F. Barnsley, V. Ervin, D. Hardin, and J. Lancaster, “Solution of an inverse problem for fractals and other sets,” Proceedings of the National Academy of Sciences of the United States of America, vol. 83, no. 7, pp. 1975–1977, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Dorigo, V. Maniezzo, and A. Colorni, “Ant system: optimization by a colony of cooperating agents,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 26, no. 1, pp. 29–41, 1996. View at Google Scholar
  16. M. Dorigo and C. Blum, “Ant colony optimization theory: a survey,” Theoretical Computer Science, vol. 344, no. 2-3, pp. 243–278, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. S. Shelokar, P. Siarry, V. K. Jayaraman, and B. D. Kulkarni, “Particle swarm and ant colony algorithms hybridized for improved continuous optimization,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 129–142, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X.-M. Hu, J. Zhang, and Y. Li, “Orthogonal methods based ant colony search for solving continuous optimization problems,” Journal of Computer Science and Technology, vol. 23, no. 1, pp. 2–18, 2008. View at Publisher · View at Google Scholar
  19. K. Socha and M. Dorigo, “Ant colony optimization for continuous domains,” European Journal of Operational Research, vol. 185, no. 3, pp. 1155–1173, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. K. Socha, “ACO for continuous and mixed-variable optimization,” M. Dorigo et al., Ed., vol. 3172 of Lecture Notes in Computer Science, pp. 25–36, Springer, Berlin, Germany, 2004. View at Google Scholar
  21. R. Chelouah and P. Siarry, “Genetic and Nelder-Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions,” European Journal of Operational Research, vol. 148, no. 2, pp. 335–348, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. R. Chelouah and P. Siarry, “A hybrid method combining continuous tabu search and Nelder-Mead simplex algorithms for the global optimization of multiminima functions,” European Journal of Operational Research, vol. 161, no. 3, pp. 636–654, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet