A ShapeNewton Approach to the Problem of Covering with Identical Balls
Abstract
The problem of covering a region of the plane with a fixed number of minimumradius identical balls is studied in the present work. An explicit construction of biLipschitz mappings is provided to model small perturbations of the union of balls. This allows us to obtain analytical expressions for first and secondorder derivatives using nonsmooth shape optimization techniques under appropriate regularity assumptions. Singular cases are also studied using asymptotic analysis. For the case of regions given by the union of disjoint convex polygons, algorithms based on Voronoi diagrams that do not rely on approximations are given to compute the derivatives. Extensive numerical experiments illustrate the capabilities and limitations of the introduced approach.
 Publication:

SIAM Journal on Scientific Computing
 Pub Date:
 April 2022
 DOI:
 10.1137/21M1426067
 arXiv:
 arXiv:2106.03641
 Bibcode:
 2022SJSC...44A.798B
 Keywords:

 Mathematics  Optimization and Control;
 49Q10;
 49J52;
 49Q12
 EPrint:
 SIAM Journal on Scientific Computing 44, pp. A798A824, 2022