A Singular Initial Value Problem to Construct Density-Equalizing Maps

Article

Abstract

The diffusion-based algorithm to produce density-equalizing maps interprets diffusion as an advection process. This algorithm uses the dynamics of a flow that is defined by an initial value problem that turns out to be very singular at the initial time. The singularities appear when the initial density has line or angle discontinuities, which is always the case, for example, in area cartogram maps. This singular initial value problem is analyzed mathematically in this article and the conclusion is that despite these singularities, it has a unique solution. This justifies the extensive numerical use of this algorithm in the recent years. The techniques presented in this article use both partial and ordinary differential equations estimates.

Keywords

Density equalizing map Cartogram Diffusion algorithm Singular initial value problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal R.P., Lakshmikantham V.: Uniqueness and nonuniqueness criteria for ordinary differential equations. World Scientific, Singapore (1993)MATHGoogle Scholar
  2. 2.
    Avinyó A., Solà-Morales J., València M. et al.: Transformations with a given Jacobian. In: Montenegro, R. (ed.) Actas XVI CEDYA, Congreso de Ecuaciones Diferenciales y Aplicaciones, VI CMA, Congreso de Matemática Aplicada. Servicio de Publicaciones y Producción Documental de la Universidad de Las Palmas de Gran Canaria, pp. 481–487. Las Palmas de Gran Canaria, Spain (1999)Google Scholar
  3. 3.
    Avinyó A., Solà-Morales J., València M.: On maps with given Jacobians involving the heat equation. Z. Angew. Math. Phys. 54(6), 919–936 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Avinyó, A., Solà-Morales, J., València, M.: On the diffusion algorithm for density-equalizing maps with piecewise constant initial data. Math. Methods Appl. Sci. (Submitted)Google Scholar
  5. 5.
    Belonosov, V.S.: Estimates of solutions of parabolic systems in weighted Hölder classes and some of their applications. Mat. Sb. 110, 163–188. English Trans: Math. USSR Sb. 38, 151–173 (1979)Google Scholar
  6. 6.
    Carrillo, J.A., Lisini, S.: On the asymptotic behavior of the gradient flow of a polyconvex functional. In: Holden, H., Karslen, K. (ed.) Nonlinear partial differential equations and hyperbolic wave phenomena, vol. 526, pp.37–51. Contemporary mathematics series AMS, Providence (2010)Google Scholar
  7. 7.
    Dacorogna B.: Direct methods in the calculus of variations. Springer-Verlag, Berlin (1998)Google Scholar
  8. 8.
    Dorling D., Newman M.E.J., Barford A.: The atlas of the real world. Mapping the way we live. Thames & Hudsons, London (2008)Google Scholar
  9. 9.
    Gastner M., Newman M.E.J.: Diffusion-based method for producing density-equalizing maps. Proc. Natl. Acad. Sci. USA. 101(20), 7499–7504 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hartman P.: Ordinary differential equations. Birkhäuser, Boston (1982)MATHGoogle Scholar
  11. 11.
    Henry D.: Geometric theory of semilinear parabolic equations, lecture notes in Math. Springer-Verlag, Berlin (1981)Google Scholar
  12. 12.
    Lekien F., Leonard N.E.: Non-uniform coverage and cartograms. SIAM. J. Control Optim. 49(1), 351–372 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mora X.: Semilinear parabolic problems define semiflows on C k spaces. Trans. Am. Math. Soc. 278, 21–55 (1983)MathSciNetMATHGoogle Scholar
  14. 14.
    Nagumo N.: Eine hinreichende Bendingung für die Unität der Lösung von Differentialgleichungen erster Ordnung. Jpn. J. Math. 3, 107–112 (1926)Google Scholar
  15. 15.
    Russo G.: Deterministic diffusion of particles. Comm. Pure Appl. Math. XLIII, 697–733 (1990)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Departament d’Informàtica i Matemàtica AplicadaUniversitat de GironaGironaSpain
  2. 2.Departament de Matemàtica Aplicada 1Universitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations