Seminario di Equazioni Differenziali e Applicazioni: Phase-field approximation of the Steiner problem

Venerdì 22 Aprile 2016, ore 11:00 - Aula 2BC30 - Filippo Santambrogio



Venerdì 22 Aprile 2016 alle ore 11:00 in Aula 2BC30, Filippo Santambrogio (Université Paris-Sud) terrà un seminario dal titolo "Phase-field approximation of the Steiner problem".

The classical Steiner problem consists in finding a connected set $K$ of minimal length containing some given points $x_0, x_1,..., x_n$. It is widely studied both in its discrete versions (in a graph, where $K$ can only be composed of edges of the graph) and in its continuous variants (in $R^d$, where all finite unions of segments are admitted). It is computationally a hard problem as $n$ increases. I will present a recent approximation result in dimension $d=2$, obtained with M. Bonnivard and A. Lemenant (Paris 7), which reminds of the Modica-Mortola or Ambrosio-Tortorelli functionals: instead of looking for a set $K$ we will look for a function $u$ and minimize a functional involving its Dirichlet energy $\int |Du|^2$ and a vanishing parameter epsilon. At the limit epsilon tends to $0$ the function $u$ will converge to the constant function $1$ a.e., expect on a very thin set ${u=0}$ which will play the role of $K$. The main difficulty will be how to handle the connectedness constraint. I will present the main tools to prove a Gamma-convergence result and the main ideas to exploit this result for numerical computations.

Rif. Int. M. Bardi, P. Mannucci, L. Caravenna.

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