Entrance hall of the ICMS

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:

  1. 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))
  2. 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.
  3. 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 July
18.30-19.00Registration & buffet, St Trinnean's Room, St Leonard's, Pollock Halls

Monday 16 July
08.30-09.00Registration
09.00-10.00Robert Douglas (University of Wales Aberystwyth)
Introduction to mass transportation and applications to geophysics
10.05-11.05Rick Salmon (Scripps Institution of Oceanography)
Mini-course on geophysical fluid dynamics (1)
11.05-11.35Coffee Break
11.35-12.20Guiseppe Buttazzo (University of Pisa)
Long-term planning versus short-term planning in location problems Presentation
12.20-14.30Lunch Break
14.30-14.45Marjolaine Puel (Université Paul Sabatier)
A mass transport approach for the relativistic heat equation Presentation
14.50-15.05Yann Brenier (CNRS Nice/Paris VI)
Polymers and optimal transportation
15.10-15.25Gordon Blower (University of Lancaster)
Young diagrams and concentration Presentation
15.25-16.05Coffee Break
16.05-16.50Wilfred Gangbo (Georgia Institute of Technology)
Euler-Poisson systems as action-minimizing paths in the Wasserstein space
16.55-17.40Eric Carlen (Georgia Institute of Technology)
Some new displacement convex functionals and some variations on displacement convexity

Tuesday 17 July
09.00-10.00Rick Salmon (Scripps Institution of Oceanography)
Mini-course on geophysical fluid dynamics (2)
10.05-10.50José Carrillo (Universitat Autònoma de Barcelona)
The Keller-Segel model: entropy, geometric inequalities and mass transport
10.50-11.20Coffee Break
11.20-12.05Aldo Pratelli (University of Pavia)
A quantitative version of the Sobolev inequality
12.10-12.25Adrian Tudorascu (Georgia Institute of Technology)
Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws Presentation
12.25-14.00Lunch Break
14.00-14.30Poster Session
14.30-14.45Bob Peeters (University of Twente)
Hamiltonian-based numerical methods for forced-dissipative climate prediction Presentation
14.50-15.05Francisco Gancedo (NRC Madrid)
Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front Presentation
15.10-15.25Maria del Mar Gonzalez (Universidad Politecnica de Catalunya)
Fractional powers of the Laplacian and the geometry of phase transition models Presentation
15.30-15.45Dorian Goldman (University of Toronto)
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing Presentation
15.50-16.05Paul Lee (University of Toronto)
Hamiltonian Reductions relating to optimal mass transport Presentation
16.05-16.35Coffee Break and Poster Session
16.35-17.20Alexander Plakhov (University of Aveiro)
Shapes of optimal aerodynamic resistance and optimal mass transportation Presentation
17.25-18.10Dejan Slepcev (Carnegie Mellon University)
Interfacial aspects of biological aggregation Presentation

Wednesday 18 July
09.00-10.00Rick Salmon (Scripps Institution of Oceanography)
Mini-course on geophysical fluid dynamics (3)
10.05-10.50Luigi Ambrosio (Scuola Normale Superiore di Pisa)
Variational models for incompressible Euler equations Presentation
10.50-11.20Coffee Break
11.20-12.05Jacques Vanneste (University of Edinburgh)
Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle Presentation
12.10-12.25Helena Lopes (University of Campinas)
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations
12.30-12.45Milton Lopes-Filho (University of Campinas)
On potential vorticity sheets for semigeostrophy
12.45-14.45Lunch Break
14.45-15.30Phil Morrison (University of Texas at Austin)
Statistical mechanics of fluid and plasma continua Presentation
15.35-16.05Coffee Break
16.05-16.50Diogo Gomez (IST Lisbon)
Linear programming and homogenization of Hamilton-Jacobi equations
16.55-17.40Marcel Oliver (Jacobs University Bremen)
Transparent boundary conditions as dissipative subgrid closures Presentation

Thursday 19 July
09.00-10.00Felix Otto (Universität Bonn)
Mini-course on some applications of optimal transportation (1)
10.05-10.50Guiseppe Savare (University of Pavia)
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds Presentation
10.50-11.20Coffee Break
11.20-11.35Philippe Delanoë (CNRS)
On the "staying away from cut-locus" issue in smooth optimal transport
11.40-12.25Gregoire Loeper (Université Claude Bernard Lyon 1/BNP Paribas)
Continuity of maps solutions of optimal transportation problems
12.30-14.35Lunch Break
14.30-15.30Felix Otto (Universität Bonn)
Mini-course on some applications of optimal transportation (2)
15.30-16.00Coffee Break
16.00-16.45Young-Heon Kim (University of Toronto)
Curvature and the continuity of optimal transportation maps Presentation
16.50-17.35Volodya Roubtsov (Université d'Angers)
Geometries and generalized geometries beyond Monge-Ampère Presentation
19.30Workshop Dinner, St Trinnean's Room, St Leonards

Friday 20 July
09.00-10.00Felix Otto (Universität Bonn)
Mini-course on some applications of optimal transportation (3)
10.05-10.50Vladimir Zeitlin (École Normale Supérieure)
Lagrangian approach to catastrophic adjustment and symmetric instability Presentation
10.50-11.20Coffee Break
11.20-12.05Boris Khesin (University of Toronto)
Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport
12.10-12.55Robert 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 Down
Blower, Gordon
Young diagrams and concentration
View Abstract Down
Brenier, Yann
Polymers and optimal transportation
View Abstract Down
Buttazzo, Guiseppe
Long-term planning versus short-term planning in location problems
View Abstract Down
Carrillo, José A
The Keller-Segel model: entropy, geometric inequalities and mass transport
View Abstract Down
Delanoë, Philippe
On the "staying away from cut-locus" issue in smooth optimal transport
View Abstract Down
Douglas, Robert
Introduction to mass transportation and applications to geophysics
View Abstract Down
Gancedo, Francisco
Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front
View Abstract Down
Gangbo, Wilfred
Euler--Poisson systems as action-minimizing paths in the Wasserstein space
View Abstract Down
Goldman, Dorian
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing
View Abstract Down
Gomes, Diogo
Linear Programming and Homogenization of Hamilton-Jacobi Equations
View Abstract Down
Khesin, Boris
Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport
View Abstract Down
Kim, Young-Heon
Curvature and the continuity of optimal transportation maps
View Abstract Down
Lee, Paul
Hamiltonian Reductions relating to optimal mass transport
View Abstract Down
Loeper, Gregoire
Continuity of maps solutions of optimal transportation problems
View Abstract Down
Lopes, Helena
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations
View Abstract Down
Lopes-Filho, Milton
On potential vorticity sheets for semigeostrophy
View Abstract Down
McCann, Robert
Ricci flow, entropy, and optimal transportation
View Abstract Down
Morrison, Phil
Statistical mechanics of fluid and plasma continua
View Abstract Down
Oliver, Marcel
Transparent boundary conditions as dissipative subgrid closures
View Abstract Down
Peeters, Bob
Hamiltonian-based numerical methods for forced-dissipative climate prediction
View Abstract Down
Plakhov, Alexander
Shapes of optimal aerodynamic resistance and optimal mass transportation
View Abstract Down
Pratelli, Aldo
A quantitative version of the Sobolev inequality
View Abstract Down
Puel, Marjolaine
A mass transport approach for the relativistic heat equation.
View Abstract Down
Salmon, Rick
Mini-course on Geophysical Fluid Dynamics
View Abstract Down
Savare, Giuseppe
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
View Abstract Down
Slepcev, Dejan
Interfacial aspects of biological aggregation
View Abstract Down
Tudorascu, Adrian
Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws
View Abstract Down
Vanneste, Jacques
Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle
View Abstract Down
Zeitlin, Vladimir
Lagrangian approach to catastrophic adjustment and symmetric instability
View Abstract Down

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