Sensitivity analysis and tailored design of minimization diagrams
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- by E. G. Birgin, A. Laurain and T. C. Menezes
- Math. Comp. 92 (2023), 2715-2768
- DOI: https://doi.org/10.1090/mcom/3839
- Published electronically: May 12, 2023
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Abstract:
Minimization diagrams encompass a large class of diagrams of interest in the literature, such as generalized Voronoi diagrams. We develop an abstract perturbation theory in two dimensions and perform a sensitivity analysis for functions depending on sets defined through intersections of smooth sublevel sets, and formulate precise conditions to avoid singular situations. This allows us to define a general framework for solving optimization problems depending on two-dimensional minimization diagrams. The particular case of Voronoi diagrams is discussed to illustrate the general theory. A variety of numerical experiments is presented. The experiments include constructing Voronoi diagrams with cells of equal size, cells satisfying conditions on the relative size of their edges or their internal angles, cells with the midpoints of pairs of Voronoi and Delaunay edges as close as possible, or cells of varying sizes governed by a given function. Overall, the experiments show that the proposed methodology allows the construction of customized Voronoi diagrams using off-the-shelf well-established optimization algorithms.References
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Bibliographic Information
- E. G. Birgin
- Affiliation: Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
- MR Author ID: 662583
- ORCID: 0000-0002-7466-7663
- Email: egbirgin@ime.usp.br
- A. Laurain
- Affiliation: Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
- MR Author ID: 764389
- ORCID: 0000-0002-8733-5190
- Email: laurain@ime.usp.br
- T. C. Menezes
- Affiliation: Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
- MR Author ID: 1376299
- ORCID: 0000-0001-5623-9545
- Email: tiagmenezes1@gmail.com
- Received by editor(s): December 15, 2021
- Received by editor(s) in revised form: October 20, 2022
- Published electronically: May 12, 2023
- Additional Notes: This work was partially supported by FAPESP (grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2021/05168-3) and CNPq (grants 302682/2019-8, 304258/2018-0, and 408175/2018-4)
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2715-2768
- MSC (2020): Primary 49Q10, 49J52, 49Q12
- DOI: https://doi.org/10.1090/mcom/3839