On the design of a reflector antenna II

Article

Abstract.

The reflector antenna design problem requires to solve a second boundary value problem for a complicated Monge-Ampére equation, for which the traditional discretization methods fail. In this paper we reduce the problem to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes an linear optimization problem and algorithms in linear programming apply.

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References

  1. 1.
    Ambrosio, L.: Lecture notes on optimal transport problems. 2000Google Scholar
  2. 2.
    Caffarelli, L., Feldman M., McCann, R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15, 1-26 (2002)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Caffarelli, L., Kochengin, S., Oliker, V.: On the numerical solution of the problem of reflector design with given far field scattering data. In: Monge-Ampere equation: application to geometry and optimization. Contemp. Math. 226, 13-32 (1999)MATHGoogle Scholar
  4. 4.
    Caffarelli, L., Oliker, V.: Weak solutions of one inverse problem in geometric optics. PreprintGoogle Scholar
  5. 5.
    Elmer, W.B.: The optical design of reflectors. Wiley, New York 1980Google Scholar
  6. 6.
    Engl, H.W., Neubauer, A.: Reflector design as an inverse problem. Proc. 5th Euro. Conf. for Mathematics in Industry, 13-24 (1991)Google Scholar
  7. 7.
    Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer. In: Current development in mathematics, pp. 65-126. International Press, Boston 1999Google Scholar
  8. 8.
    Evans L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 653, 66 (1999)MATHGoogle Scholar
  9. 9.
    Gangbo, W., McCann, R.J.: Optimal maps in Monge’s mass transport problem. C.R. Acad. Sci. Paris, Series I, Math. 321, 1653-1658 (1995)Google Scholar
  10. 10.
    Guan, P., Wang, X.-J.: On a Monge-Ampere equation arising in geometric optics. J. Diff. Geom. 48, 205-222 (1998)MathSciNetMATHGoogle Scholar
  11. 11.
    Kantorovich, L.V.: On the transfer of masses. Dokl. Akad. Nauk. USSR 37, 227-229 (1942)Google Scholar
  12. 12.
    Kantorovich, L.V.: On a problem of Monge. Uspekhi Mat. Nauk. 3, 225-226 (1948)Google Scholar
  13. 13.
    Kochengin, S., Oliker, V., von Tempski, O.: On the design of reflectors with prespecified distribution of virtual sources and intensities. Inverse Problems 14, 661-678 (1998)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Love, A.M. (eds): Reflector antennas. IEEE Press, New York 1978Google Scholar
  15. 15.
    Ma, X.N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problem. PreprintGoogle Scholar
  16. 16.
    Monge, G.: Memoire sur la Theorie des Déblais et des Remblais. Historie de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la Même année, pp. 666-704. 1781Google Scholar
  17. 17.
    Rachev, S.T., Ruschendorf, L.: Mass transportation problems. Springer, New York 1998Google Scholar
  18. 18.
    Rusch, W.V., Potter, P.D.: Analysis of reflector antennas. New York 1970Google Scholar
  19. 19.
    Trudinger, N.S., Wang, X.-J.: On the Monge mass transfer problem. Calc. Var. PDE 13, 19-31 (2001)MathSciNetMATHGoogle Scholar
  20. 20.
    Urbas, J.: Mass transfer problems. Lecture Notes, Univ. of Bonn (1998)Google Scholar
  21. 21.
    Wang, X.-J.: On the design of a reflector antenna. Inverse problems 12, 351-375 (1996)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Wang, X.-J.: The Monge optimal transportation problem. Proc. 2nd Inter. Congress Chinese Mathematicians, Taipei 2001Google Scholar
  23. 23.
    Westcott, B.S.: Shaped reflector antenna design. Research Studies Press, Letchworth 1983Google Scholar
  24. 24.
    Woods, P.J.: Reflector antenna analysis and design. Stevenage, London 1980Google Scholar
  25. 25.
    De Young, G.W.: Exploring reflection: designing light reflectors for uniform illumination. SIAM Review 42, 727-735 (2000)MATHGoogle Scholar
  26. 26.
    Khachiyan, L.G.: A polynomial algorithm for linear programming. Doklady Akademiia Nauk USSR 244, 1093-1096 (1979)MATHGoogle Scholar
  27. 27.
    Karmarkar, N.: A new polynomial time algorithm for linear programming. Combinatorica 4, 373-395 (1984)MathSciNetMATHGoogle Scholar
  28. 28.
    Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34, 250-256 (1986)MATHGoogle Scholar
  29. 29.
    Karloff, H.: Linear programming, progress in theoretical computer science. Birkhäuser, Boston 1991Google Scholar
  30. 30.
    Alevras, D., Padberg, M.: Linear optimization and extensions, problems and solutions. Springer, Berlin 2001Google Scholar

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© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Centre for Mathematics and Its ApplicationsThe Australian National UniversityCanberraAustralia

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