[Paper Review] A convex approach to the Gilbert-Steiner problem

Mauro Bonafini, Édouard Oudet|arXiv (Cornell University)|Oct 12, 2018

Markov Chains and Monte Carlo Methods24 references4 citations

TL;DR

This paper introduces a convex relaxation framework for the Gilbert–Steiner problem in Euclidean space and on manifolds, enabling numerical solution of $α$-irrigation and Steiner tree problems via a calibrated, convex variational formulation. The method achieves sharp results in many cases and enables the first numerical computation of Steiner trees on surfaces such as spheres and tori, with solutions often forming convex combinations of optimal networks.

ABSTRACT

We describe a convex relaxation for the Gilbert-Steiner problem both in $R^d$ and on manifolds, extending the framework proposed in [9], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.

Motivation & Objective

• To develop a convex relaxation for the Gilbert–Steiner problem in $\mathbb{R}^d$ and on manifolds, extending prior work on the $\alpha$-irrigation problem.
• To provide a numerically tractable, globally convergent method for minimizing the $\alpha$-irrigation energy $\int_L |\theta(x)|^\alpha d\mathcal{H}^1(x)$.
• To enable the first numerical computation of Steiner trees on curved surfaces such as spheres, tori, and punctured domains.
• To investigate the sharpness of the convex relaxation using calibration-type arguments and assess its performance on diverse geometric configurations.

Proposed method

• Formulates the $\alpha$-irrigation problem as a convex variational problem using vector fields with prescribed divergence and flux constraints.
• Employs a Raviart–Thomas finite element approach for piecewise constant vector fields and a $\mathbb{P}_2$-based method for higher-order approximations on triangulated surfaces.
• Imposes constraints on divergence and flux at source, sink, and boundary points, with tangency conditions enforced at vertices and edge midpoints.
• Uses $\Gamma$-convergence and calibration techniques to justify the sharpness of the relaxation in many cases.
• Applies adaptive mesh refinement via the Mmg platform to concentrate resolution in regions of high energy concentration.
• Solves the resulting convex optimization problem using standard numerical solvers, with solutions interpreted as convex combinations of optimal networks.

Experimental results

Research questions

• RQ1 Can a convex relaxation framework be extended to the full Gilbert–Steiner problem with multiple sources and sinks in $\mathbb{R}^d$ and on manifolds?
• RQ2 How sharp is the proposed convex relaxation for the $\alpha$-irrigation and Steiner tree problems, especially in non-Euclidean geometries?
• RQ3 Can this approach numerically compute Steiner trees on surfaces such as spheres and tori, where classical methods fail?
• RQ4 Under what geometric configurations does the convex relaxation fail to yield a convex combination of optimal trees?
• RQ5 How does the performance of the $\mathbb{P}_2$-based method compare to Raviart–Thomas in capturing energy concentration and network structure?

Key findings

• The convex relaxation successfully computes optimal networks for $\alpha$-irrigation problems in $\mathbb{R}^2$ and $\mathbb{R}^3$, with solutions often forming convex combinations of Steiner trees.
• On the sphere, the method computes the classical triple junction for three equidistant points and captures symmetric configurations with multiple minimizers.
• For four and five points on the sphere, the solution exhibits a convex combination of minimizers, with energy concentrating on the two most favorable configurations due to mesh refinement.
• On the torus, the method identifies multiple geodesic paths between antipodal points and detects non-convex-combination solutions in symmetric 3-point configurations on higher-genus tori.
• On punctured domains and surfaces with boundaries, solutions adhere to interior boundaries when energetically favorable, and non-convex combinations of optimal trees emerge, indicating potential non-sharpness.
• The $\mathbb{P}_2$-based method enables accurate energy concentration and network structure recovery on triangulated surfaces, with adaptive remeshing improving resolution in critical regions.