A Singular Initial Value Problem to Construct Density-Equalizing Maps



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.


Density equalizing map Cartogram Diffusion algorithm Singular initial value problem 


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© 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

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