Numerical approximation of the Steiner problem in dimension $2$ and $3$

Abstract

The aim of this work is to present some numerical computations of solutions of the Steiner Problem, based on the recent phase field approximations proposed in [12] and analyzed in [5, 4]. Our strategy consists in improving the regularity of the associated phase field solution by use of higherorder derivatives in the Cahn-Hilliard functional as in [6]. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.

Learning phase field mean curvature flows with neural networks

“…a Steiner tree. However, numerical experiments as proposed in [70] and in [71] show the ability of this coupling to approximate solutions of the Steiner problem in dimensions 2 and 3 but show also all the difficulties to minimize efficiently the geodesic term G ε to preserve the connectedness of the set K. The conclusions are quite similar for the approaches developed in [72,73,74] where the idea is rather to use the measure-theoretic notion of current and the connectedness of the set K is ensured by adding a divergence constraint of the form div(τ ) = α i δ a i .…”

Variational approximation of functionals defined on 1-dimensional connected sets in ℝ n

“…As pointed out in the companion paper [4], the Gilbert-Steiner problem represents the basic example of problems defined on 1-dimensional connected sets, and it has recently received a renewed attention in the Calculus of Variations community. In the last years available results focused on variational approximations of the problem mainly in the planar case [8,9,15,7], while higher dimensional approximations have been recently proposed in [10,6].…”

A convex approach to the Gilbert-Steiner problem

“…Many different variational approximations for (STP) and/or (I α ) have been proposed, starting form the simple situation where the points P i lie on the boundary of a convex set: in this case (STP) is known to be an instance of an optimal partition problem [2,3]. More recently several authors treated these problems in the spirit of Γ-convergence using approximating functionals modelled on Modica-Mortola or Ambrosio-Tortorelli type energies, initially focusing mainly on the two dimensional case [26,11,15], lately extending the same ideas also to higher dimensions [16,10].…”