Phase-field approximation for some branched transportation problems

Abstract : In this thesis we devise phase field approximations of some Branched Transportation problems. Branched Transportation is a mathematical framework for modeling supply-demand distribution networks which exhibit tree like structures. In particular the network, the supply factories and the demand location are modeled as measures and the problem is cast as a constrained optimization problem. The transport cost of a mass m along an edge with length L is h(m)xL and the total cost of a network is defined as the sum of the contribution on all its edges. The branched transportation case consists with the specific choice h(m)=|m|^α where α is a value in [0,1). The sub-additivity of the cost function ensures that transporting two masses jointly is cheaper than doing it separately. In this work we introduce various variational approximations of the branched transport optimization problem. The approximating functionals are based on a phase field representation of the network and are smoother than the original problem which allows for efficient numerical optimization methods. We introduce a family of functionals inspired by the Ambrosio and Tortorelli one to model an affine transport cost functions. This approach is firstly used to study the problem any affine cost function h in the ambient space R^2. For this case we produce a full Gamma-convergence result and correlate it with an alternate minimization procedure to obtain numerical approximations of the minimizers. We then generalize this approach to any ambient space and obtain a full Gamma-convergence result in the case of k-dimensional surfaces. In particular, we obtain a variational approximation of the Plateau problem in any dimension and co-dimension. In the last part of the thesis we propose two models for general concave cost functions. In the first one we introduce a multiphase field approach and recover any piecewise affine cost function. Finally we propose and study a family of functionals allowing to recover in the limit any concave cost function h.
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Luca Alberto Davide Ferrari. Phase-field approximation for some branched transportation problems. Optimization and Control [math.OC]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLX049⟩. ⟨tel-01914986⟩

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