Thematic period on "Calculus of Variations, Optimal Transportation, and Geometric Measure Theory: from Theory to Applications" June 27 - July 15, 2016

[Geo-Sci-Info]

Dates: June 27 - July 15, 2016

WEBSITE

Location: Université Claude Bernard Lyon 1, Lyon-Villeurbanne, France ↓ Access details

Objectives: The thematic period aims to provide an overview of the current state of research in calculus of variations, optimal transportation theory, and geometric measure theory, from both the perspectives of theory and applications. The scope of the conference ranges from rigorous mathematical analysis to modeling, numerical analysis, and scientific computing for real world applications in image processing, computer vision, physics, material science, computer graphics, biology, or data science.

Three events in three weeks :

• Week1: June 27-July 1 → First Summer School
• Week2: July 4-8 → International conference "Calculus of Variations, Geometric Measure Theory, Optimal Transportation: from Theory to Applications"
• Week3: July 11-15 → Second Summer School

Program

• First Summer School (June 27-July 1, 2016)
  Start: Monday, June 27th at 1pm
  End: Friday, July 1st at 2.30pm

  Three 7-hours lectures by

1. Dorin Bucur (U. Savoie)
   Shape optimization of spectral functionals ↓Abstract
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   In these lectures, isoperimetric type inequalities involving the spectrum of the Laplace operator (with some boundary conditions) will be seen from a shape optimisation point of view. Depending on the boundary conditions, the analysis of those problems (existence of solution, regularity, qualitative properties) is either related to a free boundary problem of Alt-Caffarelli type, or to a free discontinuity problem. I will make an introduction to this topic and present recent results, with a focus on Robin boundary conditions. In particular I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal sets.
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2. Guido De Philippis (CNRS & ENS Lyon)
   The selection principle: the use of regularity theory in proving quantitative inequalities ↓Abstract
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   I will first introduce the topic of quantitative inequalities and give some examples. Then I will present a general technique to derive them based on the regularity theory for solutions of variational problems. The course will be mainly focused on the (quantitative) isoperimetric inequality and on the (quantitative) Faber-Krahn inequality
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3. Filippo Santambrogio (U. Paris-Sud)
   Optimal transport, optimal curves, optimal flows ↓Abstract
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   The course will consist in an introduction to optimal transport theory with a special attention to the comparison between Eulerian and Lagrangian point of views, and between statical and dynamical approaches. Optimal flow versions of some issues of the problem will also be presented, and this will lead at the end of the course to the study of some traffic equilibrium problems.
   The course will consist of four lectures, roughly divided as follows:
   1 . Basic theory of Optimal Transport
   _ The problems by Monge and Kantorovich.
   _ Convex duality and Kantorovich potentials.
   _ Existence of optimal maps (Brenier Theorem) for strictly convex costs.
   2 . Wasserstein distances
   _ Definitions of the distances W_p induced by optimal transport costs.
   _ The duality between W_1 and Lipschitz functions and the topology induced by the distances W_p.
   _ The continuity equation and the curves in the space W_p.
   3 . Curves of measures and geodesics in the Wasserstein space
   _ From measures on curves to curves of measures and back.
   _ Constant-speed geodesics in the Wasserstein space.
   _ The Benamou-Brenier dynamical formulation of optimal transport.
   4 . Minimal flows
   _ An Eulerian formulation of the Monge problem with cost |x-y| (p=1): the Beckmann problem.
   _ From measures on curves to vector flows and back.
   _ Extensions to traffic congestion models.
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• Second Summer School (July 11-15, 2016)
  Start: Monday, July 11th at 9am
  End: Friday, July 15th at 6pm

  Three 7-hours lectures by

1. Daniel Cremers (Munich)
   Variational Methods for Computer Vision ↓Abstract
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   Variational methods are among the most classical and established methods to solve a multitude of problems arising in computer vision and image processing. Over the last years, they have evolved substantially, giving rise to some of the most powerful methods for optic flow estimation, image segmentation and 3D reconstruction, both in terms of accuracy and in terms of computational speed. In this tutorial, I will introduce the basic concepts of variational methods. I will then focus on problems of geometric optimization including image segmentation and 3D reconstruction. I will show how the regularization terms can be adapted to incorporate statistically learned knowledge about our world. Subsequently, I will discuss techniques of convex relaxation and functional lifting which allow to computing globally optimal or near-optimal solutions to respective energy minimization problems. Experimental results demonstrate that these spatially continuous approaches provide numerous advantages over spatially discrete (graph cut) formulations, in particular they are easily parallelized (lower runtime) and they do not suffer from metrication errors (better accuracy).
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2. Jérôme Darbon (CNRS & ENS Cachan)
   On Optimization Algorithms in Imaging Sciences and Hamilton-Jacobi equations ↓Abstract
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   The course will consist of two parts
   1. Total variation minimization and maximal flows in graphs
   Applications to image processing
   Anisotropic mean curvature flow
   2. Optimization in image processing, Hamilton-Jacobi equations, and optimal control -----------------------------------------------------------------------------------------------------------------------------_

3. Quentin Mérigot (CNRS & U. Paris-Dauphine)
   Computational optimal transport ↓Abstract
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   Optimal transport has been used as a powerful theoretical tool to study partial differential equations, differential geometry and probability for a few decades. In comparison, its use in numerical applications is much more recent, not because of lack of interest but rather because of computational difficulties. The simplest discretization of the optimal transport problem lead to combinatorial optimization problems for which can only be solved with superquadratic cost. On the other hand, the partial differential equations arising from optimal transport are fully non-linear Monge-Ampère equations, for which there did not exist robust and efficient numerical solvers until recently. This course will present a variety of approaches to solve optimal transport and related problems, with applications in mind, such as:

• Entropic penalization and Wasserstein barycenters
• Benamou-Brenier algorithm, gradient flows and simulation of non-linear diffusion equations
• Monge-Ampère equation, computational geometry and convexity constraints
• Semi-discrete optimal transport and inverse problems in geometric optics
• Measure-preserving maps, optimal quantization and Euler's equation for incompressible fluids
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• International conference "Calculus of Variations, Geometric Measure Theory, Optimal Transportation: from Theory to Applications" (July 4-8, 2016)
  Start: Monday, July 4th at 1pm
  End: Friday, July 8th at 2pm
  Conference program will be available soon

Confirmed speakers
Giovanni Alberti (Università di Pisa, Italy)
Jean-François Aujol (Université de Bordeaux, France)
Giovanni Bellettini (Università di Roma "Tor Vergata", Italy)
Virginie Bonnaillie-Noël (École Normale Supérieure de Paris, France)
Guy Bouchitté (Université du Sud-Toulon-Var, France)
Blaise Bourdin (Lousiana State University, USA)
Lia Bronsard (McMaster University, Canada)
Michael Bronstein (Università della Svizzera Italiana, Switzerland)
Almut Burchard (University of Toronto, Canada)
Daniel Cremers (Technische Universität München, Germany)
Qiang Du (Columbia University in the City of New York, USA)
Selim Esedoḡlu (University of Michigan, USA)
Ilaria Fragalà (Politecnico di Milano, Italy)
Adriana Garroni (Università di Roma "La Sapienza", Italy)
Young-Heon Kim (University of British Columbia, Canada)
Jacques-Olivier Lachaud (Université de Savoie, France)
Francesco Maggi (University of Texas at Austin, USA)
Maks Ovsjanikov (École Polytechnique, France)
Manuel Ritoré (Universidad de Granada, Spain)
Dejan Slepčev (Carnegie Mellon University, USA)
Jeremy Tyson (University of Illinois at Urbana-Champaign, USA)
Bozhidar Velichkov (Université Grenoble Alpes, France)
Max Wardetsky (Georg-August-Universität Göttingen, Germany)
Stefan Wenger (University of Fribourg, Switzerland)
Benedikt Wirth (Universität Münster, Germany)

Accommodation for PhD students and postdoctoral young researchers
Hotel accommodation (in shared double occupancy rooms) is offered to a maximal number of 60 PhD students or postdoctoral young researchers. Rooms are attributed in registration order. Use the pre-registration form below for application.

Registration fees
60 euros per week for permanent researchers
30 euros per week for PhD students and postdoctoral researchers
No registration fees for local researchers and local students
Included with conference registration fees:

• hotel accommodation for a limited number of 60 PhD students or postdoctoral young researchers (rooms are attributed in registration order)
• daily coffee and refreshment breaks
• social events
• conference dinner

Registration (deadline for payment: Monday, June 6)
NB: Week 1 being labeled as a CNRS Thematic School, it benefits from a specific funding which requires a separate registration/payment procedure. We sincerely apologize for the total inconvenience of the whole registration process.

• Step 1: If you have not done it already, fill this pre-registration form
• Step 2: For Week 1 (First Summer School, partially funded by CNRS), register and pay here→ Registration/Payment for Week #1
• Step 3: [[ Important ]] Log out from Week 1 registration process using this link
• Step 4: For Weeks 2 and/or 3 (International Conference and/or Second Summer school), register and pay here→ Registration/Payment for Weeks #2 and/or #3
• Step 5: [[ Important ]] Log out from Weeks 2-3 registration process using this link
• Step 6: Congratulations, you did it! (and many thanks for your patience...)

Scientific Committee
Lorenzo Brasco
Dorin Bucur
Antonin Chambolle
Thierry De Pauw
Guy David
Vincent Feuvrier
Antoine Lemenant
Quentin Mérigot
Benoit Merlet
Vincent Millot
Laurent Moonens
Edouard Oudet
Olivier Pantz
Séverine Rigot
Filippo Santambrogio

Organizing Committee
Elie Bretin
Sarah Delcourte
Simon Masnou
Hervé Pajot

For any questions about the thematic period, please contact us at this email address.