Article

Journal of Dynamics and Differential Equations

, Volume 24, Issue 1, pp 51-59

A Singular Initial Value Problem to Construct Density-Equalizing Maps

  • A. AvinyóAffiliated withDepartament d’Informàtica i Matemàtica Aplicada, Universitat de Girona Email author 
  • , J. Solà-MoralesAffiliated withDepartament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya
  • , M. ValènciaAffiliated withDepartament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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