Optimal Transportation, and Applications to Geophysics and Geometry
Jul 16, 2007 - Jul 20, 2007
David Hume Tower, University of Edinburgh
Organisers
Name | Institution |
---|---|
Cullen, Mike | Meteorological Office |
Scientific Organisers
Professor Luis Caffarelli, University of Texas at Austin
Dr Mike Cullen, Meteorological Office
Professor L Craig Evans, University of California, Berkeley
Professor Mikhail Feldman, University of Wisconsin-Madison
Professor Wilfred Gangbo, Georgia Institute of Technology
Professor Robert McCann, University of Toronto
Dr Ian Roulstone, University of Surrey
Short Report
The purpose of the meeting was to bring together experts in two areas of mathematics, namely mass transportation and geometry, with experts in geophysical fluid dynamics. Geophysical dynamics seeks to understand the evolution of the atmosphere and oceans, which is fundamental to weather and climate prediction. It does this by studying systems of equations which accurately model the large-scale behaviour of the solutions, rather than the small-scale turbulent motions. Mass transportation theory was originally developed as a way of solving optimisation problems in the areas of operations research, probability and statistics. It has more recently found a much wider range of applications, such as network design, economics, medical imaging and reflector design. A wide variety of other application areas were discussed at the workshop, such as biological aggregation, plasma physics, aerodynamics and relativistic physics. In particular, it has been shown that it can be applied to fluid dynamical problems, for instance those governing the large-scale behaviour of the atmosphere and oceans. Mass transportation theory can also be given a geometrical interpretation, which has led to important extensions in its applicability.
This links to fluid dynamics through the identification of incompressible flows as geodesics on an appropriate space. Recent work suggests that the classical definition of such flows has to be relaxed in order to ensure that they can be proved to exist. Recent results were described which show that mass transportation methods may be more successful in curved spaces, such as the surface of a sphere, than in at spaces. The meeting contained a significant training element with Rick Salmon giving a mini-course on geophysical fluid dynamics and Felix Otto giving a mini-course on mass transportation methods. Several talks and posters were presented by graduate students and beginning researchers.
Participants list and links to available presentations are further down this page.
Download the pdf file of the full report
Original Details
Optimal transportation is a mathematically rigorous method for solving a wide class of nonlinear optimization problems, and has been applied successfully to problems in economics, statistics, image matching, cosmology and geophysics. It has also been used to prove results in geometry and partial differential equations (PDE). A review of the method and many of the applications is given in the book by Villani (2003).
This workshop has three main themes:
- To review the links between optimal transportation and geophysical fluid dynamics. These are best developed in the context of the semi-geostrophic model, which describes large-scale flows in the atmosphere and ocean (Cullen (2006)), and in the anelastic model, which is mathematically equivalent to an incompressible model and describes small scale buoyancy-driven flows. There are also a wide variety of other models used to study the different regimes present in geophysical flows, and there are potentially new applications in these areas. (Norbury and Roulstone (2002), Majda (2003))
- To review the links between optimal transportation theory and geometry. Areas of recent progress include application of optimal transportation on manifolds, which widens the application area considerably, geometrical aspects of the theory of elliptic Monge-Ampere equations, and study of differential equations on spaces of probability measures.
- A broader survey of problems and developments surrounding optimal transportation and fluid mechanical models.
References
Ambrosio, L., Gigli, N. and Savare, G. (2005) Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lecture in Mathematics, ETH Zurich, Birkhauser.
Cullen, M. J. P. (2006) A Mathematical Theory of Large-Scale Atmospheric Flow. Imperial College Press.
Majda, A. J. (2003) Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes, 9, American Math. Society.
Norbury, J. and Roulstone, I. eds. (2002) Large-Scale Atmosphere-Ocean Dynamics., vols.1 and 2. Cambridge University Press.
Villani, C. (2003) Topics in optimal transportation. Vol. 58 of Graduate Studies in Mathematics, Amer.Math. Soc., Providence, RI.
Arrangements
Venue
The Workshop will be held in the David Hume Tower, University of Edinburgh. The Conference Rooms are equipped with OHPs, data projectors and white boards.
Registration
There will be a combined Registration and Welcome Buffet from 18.30 to 21.00 on Sunday 15 July, in the St Trinnean's Room, St Leonard's Hall, Pollock Halls. Food will be available throughout that period and you may register at any time between 18.30 and 21.00.
Those who cannot register on Sunday evening may do so at the workshop venue from 08.30-09.00 before the talks start on Monday morning.
Accommodation
ICMS has arranged en suite rooms in university accommodation in Pollock Halls for senior participants who require it, with standard accommodation for students. Participants who are being funded are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £45 per night for bed and breakfast. A list of Edinburgh accommodation of various sorts and prices can be found on the Accommodation Page on the ICMS web site. Section 4 is particularly relevant. July is very busy in Edinburgh and early booking is recommended.
Meals and Refreshments
There will be a Registration buffet in the St Trinneans Room in St Leonards, Pollock Halls, (near the Reception Centre – see map) on Sunday evening and a Workshop dinner on Thursday evening in the St Trinnean's Room. Morning and afternoon refreshments will be provided at the workshop venue.
Financial support available
We expect to be able to contribute about £100 to the travel costs of UK invited attendees, £250 for those based in Northern Europe, £350 for those based in southern Europe, £600 for those based in the USA and £750 for those based in the rest of the world.
Further information will be provided in the Invitation Letter.
Programme
Sunday 15 July18.30-19.00 | Registration & buffet, St Trinnean's Room, St Leonard's, Pollock Halls |
Monday 16 July
08.30-09.00 | Registration |
09.00-10.00 | Robert Douglas (University of Wales Aberystwyth) Introduction to mass transportation and applications to geophysics |
10.05-11.05 | Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (1) |
11.05-11.35 | Coffee Break |
11.35-12.20 | Guiseppe Buttazzo (University of Pisa) Long-term planning versus short-term planning in location problems Presentation |
12.20-14.30 | Lunch Break |
14.30-14.45 | Marjolaine Puel (Université Paul Sabatier) A mass transport approach for the relativistic heat equation Presentation |
14.50-15.05 | Yann Brenier (CNRS Nice/Paris VI) Polymers and optimal transportation |
15.10-15.25 | Gordon Blower (University of Lancaster) Young diagrams and concentration Presentation |
15.25-16.05 | Coffee Break |
16.05-16.50 | Wilfred Gangbo (Georgia Institute of Technology) Euler-Poisson systems as action-minimizing paths in the Wasserstein space |
16.55-17.40 | Eric Carlen (Georgia Institute of Technology) Some new displacement convex functionals and some variations on displacement convexity |
Tuesday 17 July
09.00-10.00 | Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (2) |
10.05-10.50 | José Carrillo (Universitat Autònoma de Barcelona) The Keller-Segel model: entropy, geometric inequalities and mass transport |
10.50-11.20 | Coffee Break |
11.20-12.05 | Aldo Pratelli (University of Pavia) A quantitative version of the Sobolev inequality |
12.10-12.25 | Adrian Tudorascu (Georgia Institute of Technology) Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws Presentation |
12.25-14.00 | Lunch Break |
14.00-14.30 | Poster Session |
14.30-14.45 | Bob Peeters (University of Twente) Hamiltonian-based numerical methods for forced-dissipative climate prediction Presentation |
14.50-15.05 | Francisco Gancedo (NRC Madrid) Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front Presentation |
15.10-15.25 | Maria del Mar Gonzalez (Universidad Politecnica de Catalunya) Fractional powers of the Laplacian and the geometry of phase transition models Presentation |
15.30-15.45 | Dorian Goldman (University of Toronto) Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing Presentation |
15.50-16.05 | Paul Lee (University of Toronto) Hamiltonian Reductions relating to optimal mass transport Presentation |
16.05-16.35 | Coffee Break and Poster Session |
16.35-17.20 | Alexander Plakhov (University of Aveiro) Shapes of optimal aerodynamic resistance and optimal mass transportation Presentation |
17.25-18.10 | Dejan Slepcev (Carnegie Mellon University) Interfacial aspects of biological aggregation Presentation |
Wednesday 18 July
09.00-10.00 | Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (3) |
10.05-10.50 | Luigi Ambrosio (Scuola Normale Superiore di Pisa) Variational models for incompressible Euler equations Presentation |
10.50-11.20 | Coffee Break |
11.20-12.05 | Jacques Vanneste (University of Edinburgh) Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle Presentation |
12.10-12.25 | Helena Lopes (University of Campinas) Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations |
12.30-12.45 | Milton Lopes-Filho (University of Campinas) On potential vorticity sheets for semigeostrophy |
12.45-14.45 | Lunch Break |
14.45-15.30 | Phil Morrison (University of Texas at Austin) Statistical mechanics of fluid and plasma continua Presentation |
15.35-16.05 | Coffee Break |
16.05-16.50 | Diogo Gomez (IST Lisbon) Linear programming and homogenization of Hamilton-Jacobi equations |
16.55-17.40 | Marcel Oliver (Jacobs University Bremen) Transparent boundary conditions as dissipative subgrid closures Presentation |
Thursday 19 July
09.00-10.00 | Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (1) |
10.05-10.50 | Guiseppe Savare (University of Pavia) Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds Presentation |
10.50-11.20 | Coffee Break |
11.20-11.35 | Philippe Delanoë (CNRS) On the "staying away from cut-locus" issue in smooth optimal transport |
11.40-12.25 | Gregoire Loeper (Université Claude Bernard Lyon 1/BNP Paribas) Continuity of maps solutions of optimal transportation problems |
12.30-14.35 | Lunch Break |
14.30-15.30 | Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (2) |
15.30-16.00 | Coffee Break |
16.00-16.45 | Young-Heon Kim (University of Toronto) Curvature and the continuity of optimal transportation maps Presentation |
16.50-17.35 | Volodya Roubtsov (Université d'Angers) Geometries and generalized geometries beyond Monge-Ampère Presentation |
19.30 | Workshop Dinner, St Trinnean's Room, St Leonards |
Friday 20 July
09.00-10.00 | Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (3) |
10.05-10.50 | Vladimir Zeitlin (École Normale Supérieure) Lagrangian approach to catastrophic adjustment and symmetric instability Presentation |
10.50-11.20 | Coffee Break |
11.20-12.05 | Boris Khesin (University of Toronto) Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport |
12.10-12.55 | Robert McCann (University of Toronto) Ricci flow, entropy, and optimal transportation |
Rick Salmon's Mini-course (NB - very large PDF file)
Felix Otto's Mini-course
Posters
Posters will be displayed all day on Tuesday 17 in the area outside the lecture theatre. The authors will be beside their posters from 14.00-14.30 and 16.05-16.35.
de Oliveira, Josiane
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations
Dubinkina, Svetlana
Statistical mechanics of Arakawa's Discretizations
Goldman, Dorian
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing
Lanagan, Gareth
Weather forecast error decomposition
Trokhimtchouk, Maxim
Optimal partition of a large labor force into working pairs
Presentations:
Presentation Details | |
---|---|
Ambrosio, Luigi | |
Variational models for incompressible Euler equations | |
View Abstract | |
Blower, Gordon | |
Young diagrams and concentration | |
View Abstract | |
Brenier, Yann | |
Polymers and optimal transportation | |
View Abstract | |
Buttazzo, Guiseppe | |
Long-term planning versus short-term planning in location problems | |
View Abstract | |
Carrillo, José A | |
The Keller-Segel model: entropy, geometric inequalities and mass transport | |
View Abstract | |
Delanoë, Philippe | |
On the "staying away from cut-locus" issue in smooth optimal transport | |
View Abstract | |
Douglas, Robert | |
Introduction to mass transportation and applications to geophysics | |
View Abstract | |
Gancedo, Francisco | |
Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front | |
View Abstract | |
Gangbo, Wilfred | |
Euler--Poisson systems as action-minimizing paths in the Wasserstein space | |
View Abstract | |
Goldman, Dorian | |
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing | |
View Abstract | |
Gomes, Diogo | |
Linear Programming and Homogenization of Hamilton-Jacobi Equations | |
View Abstract | |
Khesin, Boris | |
Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport | |
View Abstract | |
Kim, Young-Heon | |
Curvature and the continuity of optimal transportation maps | |
View Abstract | |
Lee, Paul | |
Hamiltonian Reductions relating to optimal mass transport | |
View Abstract | |
Loeper, Gregoire | |
Continuity of maps solutions of optimal transportation problems | |
View Abstract | |
Lopes, Helena | |
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations | |
View Abstract | |
Lopes-Filho, Milton | |
On potential vorticity sheets for semigeostrophy | |
View Abstract | |
McCann, Robert | |
Ricci flow, entropy, and optimal transportation | |
View Abstract | |
Morrison, Phil | |
Statistical mechanics of fluid and plasma continua | |
View Abstract | |
Oliver, Marcel | |
Transparent boundary conditions as dissipative subgrid closures | |
View Abstract | |
Peeters, Bob | |
Hamiltonian-based numerical methods for forced-dissipative climate prediction | |
View Abstract | |
Plakhov, Alexander | |
Shapes of optimal aerodynamic resistance and optimal mass transportation | |
View Abstract | |
Pratelli, Aldo | |
A quantitative version of the Sobolev inequality | |
View Abstract | |
Puel, Marjolaine | |
A mass transport approach for the relativistic heat equation. | |
View Abstract | |
Salmon, Rick | |
Mini-course on Geophysical Fluid Dynamics | |
View Abstract | |
Savare, Giuseppe | |
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds | |
View Abstract | |
Slepcev, Dejan | |
Interfacial aspects of biological aggregation | |
View Abstract | |
Tudorascu, Adrian | |
Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws | |
View Abstract | |
Vanneste, Jacques | |
Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle | |
View Abstract | |
Zeitlin, Vladimir | |
Lagrangian approach to catastrophic adjustment and symmetric instability | |
View Abstract |
Participants
Name | Institution |
---|---|
Ambrosio, Luigi | Scuola Normale Superiore di Pisa |
Barrett, John | Imperial College, London |
Bevan, Jonathan | University of Surrey |
Blower, Gordon | University of Lancaster |
Brenier, Yann | CNRS Nice |
Buffoni, Boris | EPF Lausanne |
Burton, Geoffrey | University of Bath |
Buttazzo, Guiseppe | University of Pisa |
Carlen, Eric | Rutgers University |
Carrillo, José A | ICREA & Universitat Autònoma de Barcelona |
Carvalho, Maria | University of Lisbon |
Castro, Angel | CSIC Madrid |
Chen, Jun | University of Wisconsin |
Cordoba, Diego | NRC Madrid |
Cullen, Mike | Meteorological Office |
de Oliveira, Josiane Cristina | UNICAMP |
Delahaies, Sylvain | University of Surrey |
Delanoë, Philippe | CNRS |
Douglas, Robert | Aberystwyth University |
Dubinkina, Svetlana | CWI Amsterdam |
Feldman, Mikhail | University of Wisconsin-Madison |
Figalli, Alessio | Scuola Normale Superiore di Pisa |
Gancedo, Francisco | NRC Madrid |
Gangbo, Wilfred | Georgia Institute of Technology |
Goldman, Dorian | University of Toronto |
Gomes, Diogo | IST Lisbon |
Gonzalez, Maria del Mar | University of Texas at Austin |
Karakhanyan, Aram | University of Edinburgh |
Khesin, Boris | University of Toronto |
Kim, Hwa Kil | Georgia Institute of Technology |
Kim, Young-Heon | University of Toronto |
Lanagan, Gareth | University of Aberystwyth |
Lee, Paul | University of Toronto |
Lisini, Stefano | Universita del Piemonte Orientale |
Loeper, Gregoire | Université Claude Bernard Lyon 1/BNP Paribas |
Lopes, Helena | University of Campinas |
Lopes-Filho, Milton | University of Campinas |
Maggi, Francesco | Università degli Studi di Firenze |
Markowich, Peter | University of Cambridge |
McCann, Robert | University of Toronto |
Morrison, Phil | University of Texas at Austin |
Norbury, John | University of Oxford |
Oliver, Marcel | Jacobs University Bremen |
Otto, Felix | Universität Bonn |
Pacini, Tommaso | Scuola Normale Superiore, Pisa |
Panike, Nathan | University of Wisconsin |
Peeters, Bob | University of Twente |
Plakhov, Alexander | University of Aveiro |
Pratelli, Aldo | University of Pavia |
Puel, Marjolaine | Université Paul Sabatier |
Roubtsov, Volodya | Université d'Angers |
Roulstone, Ian | University of Surrey |
Salmon, Rick | Scripps Institution of Oceanography |
Savare, Giuseppe | University of Pavia |
Schonbek, Maria | University of California Santa Cruz |
Slepcev, Dejan | Carnegie Mellon University |
Sosio, Maria | University of Toronto |
Spirn, Dan | University of Minnesota |
Thuburn, John | University of Exeter |
Trokhimtchouk, Maxim | University of California, Berkeley |
Tudorascu, Adrian | Georgia Institute of Technology |
Vanneste, Jacques | University of Edinburgh |
Wirosoetisno, Djoko | University of Durham |
Yolcu, Turkay | Georgia Institute of Technology |
Zeitlin, Vladimir | École Normale Supérieure |