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Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

Organizers: José Carrillo (ICREA, UAB Barcelona), Marco Di Francesco (U. of L’Aquila), Yann Brenier (CNRS Nice), Peter A. Markowich (WPI c/o U.Vienna), Walter Schachermayer (WPI c/o TU Vienna), Juan Luis Vazquez (UAM Madrid)

General description of the program.
The “optimal transportation problem” (originally going back to Monge) was revived in the mid eighties by the work of Yann Brenier, who characterized the optimal transfer plans in terms of gradients of convex functions. In the last decades, this problem has been recovered to have a close relationship with certain evolutionary PDE's, which can be interpreted as gradient flows of certain entropy functionals with respect to a metric (well-known to probabilists, see e.g. Knott-Smitt, Rachev-Rueschendorf) involving optimal transportation called Wasserstein metric. The first application to mathematical physics (kinetic models) is due to Tanaka, in the seventies. In the early nineties the use of entropy functionals as a tool to prove convergence to equilibrium received a strong impulse due to the work of Cercignani, Carlen, Carvalho, Pulvirenti, Desvillettes, Toscani, Villani and others. Moreover, Toscani proved that similar methods could be used to prove optimal convergence to similarity for diffusion equations. At the same time Jordan, Kinderlehrer and Otto discovered that the Fokker Planck equation can be solved by a steepest descent method involving a logarithmic entropy functional and the Wasserstein distance. This work marks the beginning of the modern gradient flow theory on Wasserstein spaces. After a few years, Arnold, Carrillo, Del Pino, Dolbeault, Jüngel, Markowich, Toscani and Unterreiter established the link between convergence to equilibrium for linear and nonlinear Fokker-Planck type equations and logarithmic Sobolev inequalities, by developing a previous idea of Bakry and Emery (with applications to the Porous medium equation). The key ingredients of this theory (the log-Sobolev and the Csiszar-Kullback inequalities) are related to certain Gaussian isoperimetric inequalities (see e.g. Talagrand and Otto-Villani).Otto realized simultaneously that nonlinear diffusion equations can be seen as gradient flows in the 2-Wasserstein space of probability measures of a free energy functional. This metric structure has been made rigorous by Ambrosio-Gigli-Savarè. At the same time, Carrillo-McCann-Villani applied these ideas to granular media models producing these arguments in smooth settings. A basic ingredient of this theory is the notion of convexity along geodesics in the Wasserstein space introduced by McCann, also called “displacement convexity”. Further applications of this theory are to space inhomogeneous Boltzmann equations (Desvillettes-Villani), Drift diffusion models in semiconductors (Arnold-Biler-Dolbeault-Markowich-Toscani) Fourth order equations (Caceres-Carrillo-Toscani), Reaction diffusion systems (Desvillettes-Fellner), Fast diffusion equations (Markowich-Lederman, Carrillo-Vazquez), Doubly degenerate parabolic equations (Agueh), Quantum-drift-diffusion models (Gianazza-Savarè-Toscani), Keller-Segel type models for chemotaxis (Blanchet-Dolbeault-Perthame). Another striking application of the optimal transportation (from the probabilistic point of view based on martingales theory) is the justification of the mean field limits of certain stochastic particle models by means of the theory of concentration inequalities developed (among the others) by Lévy, Gromov, Milman, Bobkov, Ledoux, Malrieu. A computational method for finding entropy functionals for evolutionary equations has been recently proposed by Juengel-Mattes. The use of the Wasserstein distance has been also extended to scalar conservation laws (Bolley-Brenier-Loeper, Carrillo-Di Francesco-Lattanzio). The use of these ideas to study the long-time asymptotics of dissipative homogeneous kinetic models is based on the almost equivalence of the Euclidean transportation metric with Fourier-based metrics and on the basic mechanism of contraction of probability metrics (Gabetta-Toscani-Wennberg, Bisi-Carrillo-Toscani, Bolley-Carrillo, Carrillo-Toscani). Several (important) authors have been involved in literature of the optimal transportation theory, with remarkable applications, we mention here Cullen, Caffarelli, Urbas, Evans, Barthe, Ghoussoub, Lott, Carlen and Gangbo.


Scientific goals:

  • The point of view of optimal transportation and the use of the techniques related with it have been proven to be easily implemented to a wider and wider range of fields in the applied mathematics. Our main goal is to explore possible applications to further fields like compressible fluid mechanics and magneto-hydrodynamics, image processing and Cahn-Hilliard type equations, quantum hydro-dynamics models, cells migration models from mathematical biology, non-local particle aggregation models.
  • Image processing is starting to use concepts of measure transport for different applications. Several authors (e. g. Chan, Shen, Bertozzi) proposed the use of the Cahn-Hilliard equation in the context of image inpainting. The gradient flow structure of the Cahn-Hilliard equation involves certain interesting multiscale issues. Moreover, a rigorous justification of the gradient flow structure by means of optimal transportation is still open, especially due to the lack of convexity of the functional in the gradient flow structure. A more direct application of optimal transportation to image processing is due to Sigurd Angenent, Alan Tannenbaum and collaborators. They showed that algorithms providing matching correspondence between images (warping or interpolation in registration) of a film can be devised by using optimal transportation. Several applications to biomedical images has been produced by this group using similar techniques. Last but no least, Caselles, Bernot and Morel have introduced a new concept of irrigation measures which can be considered a more sophisticated and better adapted to image processing than optimal transport of measures. A healthy interaction of these people with the persons already working in optimal transport would be beneficial for both side opening and widening both the theoretical knowledge and the applications fields of the theory.
  • The application of optimal transportation theory to Keller-Segel type models for chemotaxis is at its beginning. The recent result of Blanchet-Dolbeault-Perthame pinpointed the problem of giving a gradient flow structure to these models, a challenging problem due the blow up phenomena occurring in several cases. Several recent results include the understanding of the critical case by Blanchet-Carrillo-Masmoudi and the convergence of the Jordan-Kinderlehrer-Otto scheme for the subcritical Keller-Segel model by Blanchet-Calvez-Carrillo. To attack issues related to the asymptotic behaviour of solutions for gradient flows of bounded from below not convex energy functionals is an important open problem as it can be seen from particular instances as the sub-critical Keller-Segel model in 2D.
  • An important goal is the interdisciplinary communication between the experts in PDE’s and the experts in probability theory, in order to foster a shared point of view, especially concerning mean field limits of particle models and the link between Gaussian isoperimetry and optimal transportation inequalities. Moreover, the connections to the limit of particle systems governed by SDEs has simplified several proofs of contractions for granular media models (Cattiaux-Guillin-Malrieu). In fact, probability tool will be helpful to attack several open problems as convergence of particle systems towards continuum PDE models.
  • Recently, models enjoying a formal gradient flow structure with a nonlinear mobility have been proposed in biomathematical models and in thin film dynamics. A general framework in the spirit of Ambrosio-Gigli-Savarè, or Carrillo-McCann-Villani is still missing, even though various ideas have been shared.
  • Models including nonlinear convection have been partially studied in this context. Several collaborations (Carrillo-Fellner, Carrillo-Di Francesco-Lattanzio, Bolley-Brenier-Loeper) have produced interesting results, but the theory is still incomplete. This program could be the occasion to interact with experts in gradient flows and semigroup theory like e.g. Savarè and Vazquez.
  • Events

    Workshop "Optimal transportation structures, gradient flows and entropy methods for applied PDE's"

    Location: Seminarroom C206 + C207
    Time: 24. Sep 2007 (Mon) - 26. Sep 2007 (Wed); Opening: 9:00
    Organisation(s)
    WPI
    Organiser(s)
    José A. Carrillo (UAB Barcelona)
    Marco Di Francesco (U. L'Aquila)
    Remark: Opening: Monday, September 24, h 9.15.

    Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's"

    Location: Seminarroom C206 + C207
    Time: 10. Sep 2007 (Mon) - 21. Sep 2007 (Fri); Opening: 9:00
    Organisation(s)
    WPI
    Organiser(s)
    José A. Carrillo (UAB Barcelona)
    Marco Di Francesco (Univ. L'Aquila)

    Minicourse on "Branched transport theory"
    Speakers: Jean Michel Morel and Filippo Santambrogio (ENS Cachan)

    Location: WPI Seminarroom C714
    Time: 18. Jun 2007 (Mon) - 20. Jun 2007 (Wed); Opening: 9:50
    Organisation(s)
    Wolfgang Pauli Institut
    Organiser(s)
    Marco Di Francesco (U. L'Aquila)
    Remark: Click HERE for lecture notes.

    Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"
    Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna)

    Location: WPI Seminarroom C714
    Time: 11. Jun 2007 (Mon) - 22. Jun 2007 (Fri); Opening: 9:00
    Organisation(s)
    Wolfgang Pauli Institut
    Organiser(s)
    Marco Di Francesco (Univ. L'Aquila)

    Mini-workshop on "PDE's and Variational Tools in Image Inpainting"

    Location: WPI Seminarroom C714
    Time: 11. Jun 2007 (Mon) - 12. Jun 2007 (Tue); Opening: 9:45
    Organisation(s)
    Wolfgang Pauli Institut
    Organiser(s)
    Peter A. Markowich
    Marco Di Francesco
    Massimo Fornasier

    Introductory minicourses on "Optimal Transportation, gradient flows and entropy methods"

    Location: WPI, Seminarroom C714
    Time: 7. May 2007 (Mon) - 16. May 2007 (Wed); Opening: 9:00
    Organisation(s)
    Wolfgang Pauli Institut
    Organiser(s)
    Josef Teichmann (TU Vienna)
    Klemens Fellner (Univ. of Vienna)
    Marco Di Francesco (Univ. L'Aquila)

    Talks in the framework of this thematic program... (by date) , (by name)

    Pauli Fellows

    Brenier, Yann 1. Aug 2007-30. Sep 2007 local address 
    Vazquez, Juan Luis 1. Sep 2007-30. Sep 2007 local address 

    Visitors

    Agueh, Martial 23. Sep 2007-29. Sep 2007 local address 
    Antonelli, Paolo 12. Jun 2007-21. Jun 2007 local address 
    Apushkinskaya, Darya 10. Jun 2007-13. Jun 2007 local address 
    Biler, Piotr 23. Sep 2007-26. Sep 2007 local address 
    Bolley, Francois 23. Sep 2007-29. Sep 2007 local address 
    Bonforte, Matteo 6. Aug 2007-9. Aug 2007 local address 
    Burger, Martin 6. Aug 2007-10. Aug 2007 local address 
    Carrillo, José A. 6. Aug 2007-10. Aug 2007 local address 
    16. Sep 2007-29. Sep 2007 local address 
    Cuadrado, Silvia 23. Sep 2007-25. Sep 2007 local address 
    Figalli, Alessio 8. Sep 2007-25. Sep 2007 local address 
    Fornasier, Massimo 10. Jun 2007-13. Jun 2007 local address 
    Gerasymenko, Viktor 2. May 2007-11. May 2007 local address 
    Grossauer, Harald 10. Jun 2007-13. Jun 2007 local address 
    Gualdani, Maria Pia 11. Jun 2007-22. Jun 2007 local address 
    Guillin, Arnaud 23. Sep 2007-26. Sep 2007 local address 
    Kuijper, Arjan 10. Jun 2007-13. Jun 2007 local address 
    Liu, Hailiang 27. May 2007-3. Jun 2007 local address 
    Malrieu, Florent 23. Sep 2007-26. Sep 2007 local address 
    Manzini, Chiara 15. Sep 2007-29. Sep 2007 local address 
    Matthes, Daniel 12. Jun 2007-20. Jun 2007 local address 
    23. Sep 2007-26. Sep 2007 local address 
    Moll, Salvador 23. Sep 2007-27. Sep 2007 local address 
    Morel, Jean-Michel 16. Jun 2007-20. Jun 2007 local address 
    Ni, Kanguy 10. Jun 2007-13. Jun 2007 local address 
    Puel, Marjolaine 20. Sep 2007-26. Sep 2007 local address 
    Rosado Linares, Jesus 10. Jun 2007-20. Jun 2007 local address 
    9. Sep 2007-27. Sep 2007 local address 
    Santambrogio, Filippo 16. Jun 2007-20. Jun 2007 local address 
    Savaré, Giuseppe 13. Jun 2007-21. Jun 2007 local address 
    Simeoni, Chiara 16. Sep 2007-26. Sep 2007 local address 
    Toscani, Giuseppe 17. Sep 2007-20. Sep 2007 local address 
    Wolfram, Marie T. 8. Aug 2007-10. Aug 2007 local address 

    PostDocs

    Blanchet, Adrien WK 1. Jul 2007-30. Sep 2007 local address 
    Bonforte, Matteo 1. Sep 2007-30. Sep 2007 local address 
    DiFrancesco, Marco 1. May 2007-31. Jul 2007 local address 
    Lisini, Stefano 1. Sep 2007-30. Sep 2007 local address 

    PreDocs

    Antonelli, Paolo 1. Sep 2007-30. Sep 2007 local address 
    Calvez, Vincent 1. Sep 2007-30. Sep 2007 local address 
    Caravenna, Laura 1. Sep 2007-30. Sep 2007 local address 
    Donadello, Carlotta 1. Sep 2007-30. Sep 2007 local address 
    Gloyer, Matteo 1. Sep 2007-30. Sep 2007 local address 
    Natile, Luca 1. Sep 2007-30. Sep 2007 local address 
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