On the diffusion algorithm for density‐equalizing maps with piecewise constant initial data

Abstract

We mathematically analyze the diffusion-based algorithm to produce maps with a given Jacobian, introduced independently\ud by M. T. Gastner and M. E. J. Newman (2004) and ourselves (2003), but in particular cases where the initial density\ud has line or angle discontinuities in the plane. In this situation, the conclusion reinforces the conjecture that the algorithm\ud is always well-posed, in accordance with its extensive numerical use in some areas of applied sciences (cartograms, sensor\ud networks, computational grids, or image registration).Peer ReviewedPostprint (published version

A Singular Initial Value Problem to Construct Density-Equalizing Maps

“…We concentrate on the most difficult part, namely uniqueness. Some particular and simpler cases were studied in our previous publication [4]. The previous calculations are the origin of this work in the sense that gave us confidence on the fact that the effect of these singularities would not be to break the uniqueness of solutions.…”