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Article Dans Une Revue ESAIM: Mathematical Modelling and Numerical Analysis Année : 2024

On the numerical approximation of Blaschke-Santalo diagrams using Centroidal Voronoi Tessellations

Giuseppe Buttazzo
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Edouard Oudet

Résumé

Identifying Blaschke-Santal\'o diagrams is an important topic that essentially consists in determining the image $Y=F(X)$ of a map $F:X\to\mathbb{R}^d$, where the dimension of the source space $X$ is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, $X$ can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number $N$ of points $x_1,\dots,x_N$ in $X$ and plotting them in the target space $\mathbb{R}^d$, produces in many cases areas in $Y$ of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points $x_i$ in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set $Y$ by a relatively small number $N$ of samples which is very useful when the dimension of the source space $X$ is large (or even infinite) and the evaluation of $F(x_i)$ is costly. Our method consists in a suitable use of {Centroidal Voronoi Tessellations} which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.
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Dates et versions

hal-03966754 , version 1 (31-01-2023)

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Citer

Beniamin Bogosel, Giuseppe Buttazzo, Edouard Oudet. On the numerical approximation of Blaschke-Santalo diagrams using Centroidal Voronoi Tessellations. ESAIM: Mathematical Modelling and Numerical Analysis, 2024, 58 (1), pp.393-420. ⟨10.1051/m2an/2023092⟩. ⟨hal-03966754⟩
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