Volume 5, Issue 2
Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 242-259.

Published online: 2012-05

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• Abstract

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and $O(k)$ complexity estimation is provided for a problem with $k$ generators.

• Keywords

Centroidal Voronoi tessellation, optimal quantization, truncated Newton method, Lloyd's algorithm, multilevel method, uniform convergence.

15A12, 65F10, 65F15

• BibTex
• RIS
• TXT
@Article{NMTMA-5-242, author = {}, title = {Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {2}, pages = {242--259}, abstract = {

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and $O(k)$ complexity estimation is provided for a problem with $k$ generators.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1046}, url = {http://global-sci.org/intro/article_detail/nmtma/5937.html} }
TY - JOUR T1 - Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 242 EP - 259 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1046 UR - https://global-sci.org/intro/article_detail/nmtma/5937.html KW - Centroidal Voronoi tessellation, optimal quantization, truncated Newton method, Lloyd's algorithm, multilevel method, uniform convergence. AB -

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and $O(k)$ complexity estimation is provided for a problem with $k$ generators.

Zichao Di, Maria Emelianenko & Stephen Nash. (2020). Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation. Numerical Mathematics: Theory, Methods and Applications. 5 (2). 242-259. doi:10.4208/nmtma.2012.m1046
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