Filippo Santambrogio
Prof. Dr. Filippo Santambrogio

Technische Universität München
Zentrum Mathematik, M8
Boltzmannstraße 3
85748 Garching bei München
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Raum: 02.08.035

Homepage: Filippo Santambrogio

Optimal Networks and Branched Transport

Preview: The course will be devoted to various problems where the unknown is a network connecting, or getting close to, given points, to be chosen so that a certain cost, depending on its length and possibly on a suitable notion of weight, is minimized. This shows straightforward engineering applications and has connections with many interesting parts of mathematics, in particular in optimal transport, calculus of variations and geometric measure theory. We will in particular focus on the Steiner problem (finding a connected set of minimal length, forced to include some given points) and on the so-called irrigation problem (finding a connected set of prescribed length, minimizing the average distance of a given distribution of mass to it), and prove the Golab Theorem which is the key for the existence theory: the length is a lower-semicontinuous functional for the Hausdorff convergence in the class of closed connected sets. We will then move to a different class of problems, called branched transport, where the network will also feature a weight, representing the amount of mass travelling on it, forced to satisfy Kirchhoff's balance law. When the cost is subadditive with respect to this weight (transporting twice the mass costs more, but now twice as much) then optimal networks feature branched structures that can be found in many human-built systems, as well as in nature (plants, roots, rivers,...). This is a variant of optimal transport problems, for which some equivalent of the Kantorovich potentials (the so-called landscape function, whose introduction is motivated by geophysical models about river basins) and of the Wasserstein distances can be introduced. The last part of the course will be devoted to phase-field approximation of these network problems: if one wants to numerically approach the optimization it is better to optimize among functions instead of among sets, minimizing an approximated functional depending on a small parameter ε which will correspond to the size of a thickened network (represented in many case by a level set of the corresponding function).

Contents: {{Versteckt|Dieser Bereich wird noch aktualisiert.
  • 1. Chapter:
    Introduction to network optimization problems
    The Steiner problem, the irrigation problem, and branched transport, together with their angle laws. In all these problems the unknown is a 1D network and the cost and the constraints involve its length and its suitability to reach a given target of points.
  • 2. Chapter:
    Hausdorff measures
    Some bases about positive measures and their convergence. Definition of k-dimensional Hausdorff measures. Densities and covering theorems.
  • 3. Chapter:
    Hausdorff convergence and Golab's Theorem
    The notion of Hausdorff convergence on the set of closed sets. Proof of the fact that the H^1 measure is l.s.c. for this convergence in the class of connected sets.
  • 4. Chapter:
    Existence and main properties for the optimal sets in the Steiner and irrigation problems
    Application of the Golab's Theorem. Absence of loops. Some ideas about the blow-up of optimal sets around their points and about their regularity.
  • 5. Chapter:
    Branched transport: the Eulerian point of view
    Xia's definition as an optimal flow problem under divergence constraints with a cost proportional to the mass to the power α. Finiteness of the distance if α>1-1/d.
  • 6. Chapter:
    Comparison with standard optimal transport and between branched transport distances and Wasserstein distances. Alternative approaches
    The Beckmann's minimal flow problem for the Monge cost and the distance W_1. Quantitative comparison between branched transport distances and W_1. A relaxed Kantorovich formulation (Maddalena-Solimini) and a Benamou-Brenier one. Comparison with W_{1/α}.
  • 7. Chapter:
    Branched transport: the Lagrangian point of view
    Minimization among measures on curves (Maddalena-Morel-Solimini). Various definitions of multiplicities. The case of a single source.
  • 8. Chapter:
    The landscape function
    Geophysical models for river basins and erosion. Optimal channel networks and landscape functions in a discrete setting. Definition in a continuous setting and comparison with the Kantorovich potential.
  • 9. Chapter:
    Properties and applications of the landscape function
    Lower semicontinuity. Holder continuity under lower bounds on the target measure. Maximal Slope in the direction of the network. Application to variational problems, shape optimization.
  • 10. Chapter:
    Introduction to phase-field models and Γ-convergence
    Basic definitions and properties of Γ-convergence. The example of Modica-Mortola as an approximation of the perimeter. Application to the Steiner problems to connect points on the boundary of a 2D convex polygon
  • 11. Chapter:
    A Modica-Mortola like approximation of branched transport problems.
    Minimization of a convex-concave energy among smooth vector fields under divergence constraints: Γ-convergence
  • 12. Chapter:
    A phase-field approximation for the Steiner problem.
    Weighted distance functions. Minimization of a Modica-Mortola term with weighted distance penalization. Convergence to a minimizer.
  • ECTS Credits:

    MA 5925 Geometric Measure Theory and Applications
    MA 2501 Algorithmische Diskrete Mathematik

    Study goals:
    After successful completion of the module students are able to model classical transport or network problems as a rigorous variational problem and to study the mathematical questions that they bring (existence, regularity, approximation...) by handling the main relevant analysis tools.

    Teaching meethods:
    The module is offered as lectures. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Discussion sessions can be organized if students are interested.

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