A PDE approach to centroidal tessellations of domains

We introduce a class of systems of Hamilton-Jacobi equations that characterize critical points of functionals associated to centroidal tessellations of domains, i.e. tessellations where generators and centroids coincide, such as centroidal Voronoi tessellations and centroidal power diagrams. An appropriate version of the Lloyd algorithm, combined with a Fast Marching method on unstructured grids for the Hamilton-Jacobi equation, allows computing the solution of the system. We propose various numerical examples to illustrate the features of the technique.