Oct 14, 2013
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Optimal transport and curvature -- theorems and problems
VILLANI Cedric (Institute Henri Poincare)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Lagrangian solutions of semigeostrophic system with singular initial data
Mikhail Feldman (University of Wisconsin-Madison)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- We discuss some generalizations of weak Lagrangian solutions for the semigeostrophic
system in physical space, to include the possibility of singular measures in the dual space. We
discuss existence of such solutions, and their properties, including the link with the solutions
in dual space, and conservation of geostrophic energy. The talk is based on joint work
with A. Tudorascu.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Existence of distributional solutions to the semigeostrophic equations
Maria Colombo (Scuola Normale Superiore)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The semigeostrophic equations are a set of equations which model large-scale atmospheric/ocean flows. The system admits a dual version, obtained from the original equations through a change of variable. Existence for the dual problem has been proven in 1998 by Benamou and Brenier, but the existence of a solution of the original system remained open due to the low regularity of the change of variable. In the talk we prove the existence of distributional solutions of the original equations, both in R^3 and in a two-dimensional periodic setting. The proof is based on recent regularity and stability estimates for Alexandrov solutions of the Monge-Amp`ere equation, established by De Philippis and Figalli.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Level set volume preserving diffusions
Yann Brenier (École Polytechnique)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- We discuss diffusion equations that are constrained to preserve the volume
of each level set during the time evolution (which excludes the standard
heat equation). We consider, in particular, the gradient flow of the
Dirichlet integral under suitable volume-preserving transportation metrics.
The resulting equations are non-linear and very degenerate, admitting as
stationary solutions all scalar functions which are functions of their own
Laplacian. (In particular, in 2D, all stationary solutions of the Euler
equations for incompressible fluids.) We relate them to both combinatorial
optimization and linear algebra, through the quadratic assignmemt problem (a
NP combinatorial optimization problem including the travelling salesman
problem) and the Brockett-Wegner diagonalizing flow for linear operators.
For these equations, we provide a concept of "dissipative solutions" that
exist globally in time and are unique as long as they stay smooth, following
some works of P.-L. Lions (for the Euler equations) and
Ambrosio-Gigli-Savare (for the heat equation in metric spaces).
- Supplements
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Oct 15, 2013
Tuesday
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09:30 AM - 10:30 AM
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Optimal Mass Transport in Medical Imaging Computation
Allen Tannenbaum (State University of New York, Stony Brook)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- Optimal mass transport methods have recently become very important for various problems in medical imaging analysis including registration and anatomical shape. They have been included in software packages, e.g., the 3D Slicer of the Harvard Medical School. We will describe some of the key issues in medical imaging, and how optimal mass transport can be used to shed some light on the solution of these problems. Applications include left atrial fibrillation, traumatic brain injury, and tumor growth models. Very fast implementations using GPUs, even make these methods suitable in an intraoperative setting
- Supplements
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Fokker-Planck equations, Free Energy, and Markov Processes on Graphs
Haomin Zhou (Georgia Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The classical Fokker-Planck equation is a linear parabolic equation which describes the time evolution of probability density of a stochastic process defined on an Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and entropy. In recent years, it has been shown that the Fokker-Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this talk, we present results on similar matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If $N\ge 2$ is the number of vertices of the graph, we show that the corresponding Fokker-Planck equation is a system of $N$ {\it nonlinear} ordinary differential equations defined on a Riemannian manifold of probability distributions.
However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker-Planck equations for the same process. It is shown that there is a strong connection but also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker-Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples will also be discussed. The work is jointly with Shui-Nee Chow (Georgia Tech), Wen Huang (USTC) and Yao Li (Courant Institute).
- Supplements
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Zhou
2.31 MB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Imaging of flow in porous media from reconstruction to prediction
Eldad Haber (University of British Columbia)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- In this talk we discuss an inverse problem where we aim to recover fluid that is transported in porous media.
We discuss a joint inversion for the flow and the fluid saturation and show that by including the transport process within the reconstruction we are able to predict the flow.
- Supplements
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Haber
3.3 MB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Convexity constraint and related problems
Edouard Oudet (Université de Grenoble I (Joseph Fourier))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- We discuss different discretization of convexity constraint which have been introduced last decades. This natural question is strongly related with W2 optimal transportation and several unsolved geometrical problems
- Supplements
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Oudet
1.68 MB application/pdf
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Supplements
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Oct 16, 2013
Wednesday
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09:30 AM - 10:30 AM
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Free upper boundary value problems for the semi-geostrophic equations
Michael Cullen (Met Office)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- I consider the flow of three-dimensional stratified rotating fluid with a free upper boundary. I show how the semi-geostrophic equations are derived as a limit of the Euler equations. Following earlier work of Benamou and Brnier, and Cullen and Gangbo, the equations are formulated in dual variables and the mapping to physical space is determined by optimal transportation using the energy as the cost function. I focus on the differences from the earlier work. These are the form of the energy, the definition of the space in which the energy is to be minimised, and the proof that the energy is strictly convex with respect to variations in the free upper boundary
- Supplements
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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A PDE approach to computing viscosity solutions of the Monge-Kantorovich problem
Jean David Benamou (Institut National de Recherche en Informatique Automatique (INRIA))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- I will present a new technique to deal with the state constraint that binds the transport when source and target have
compact support. It takes the form of non-linear boundary conditions which can be combined to a Monge-Ampère equation to
solve the optimal transport problem. The wide-stencil discretization technique and fast Newton solver proposed by Oberman and Froese
is extended to this framework and allows to compute weak viscosity solution of the optimal transport problem.
Numerical solutions will be presented to illustrate strengths and weaknesses of the method.
- Supplements
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Oct 17, 2013
Thursday
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09:30 AM - 10:30 AM
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Blowup Dynamics for Nonlocal Transport Problems
Andrea Bertozzi (University of California, Los Angeles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- In this talk I consider the dynamic problem of finite time blowup in nonlocal equations. I will review work involving blowup in nonlocal aggregation equations and how it relates to classical problems in fluids. The talk will focus on numerical simulations and models. I will also discuss some recent work on contagion models for crowd dynamics. I will discuss both kinetic and macroscopic models for such problems and discuss the connection of this problem to the classical "sticky particle" model in gas dynamics.
- Supplements
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Standard Finite Elements for the Numerical Resolution of the Elliptic Monge-Ampère Equation
Gerard Awanou (University of Illinois at Chicago)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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Awanou
98.9 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Simulations of Fluvial Landscapes and Optimal Transport
Björn Birnir (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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Birnir
42.2 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Density functional theory and optimal transportation with Coulomb cost
Gero Friesecke (TU München)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- Density functional theory (DFT) is a computationally feasible
electronic structure model which simplifies full quantum mechanics and
for which Walter Kohn received a Nobel prize in 1998. In the
semiclassical limit, DFT reduces to a multi-marginal optimal transport
problem [1]. Considerable insight into the limit problem had been built
up, prior to our work, by physicists (Seidl, Perdew, Levy, Gori-Giorgi,
Savin), who essentially developed a considerable amount of optimal
transport theory without knowing they were doing optimal transport. The
goal of my talk is three-fold
(i) to explain the connection DFT--optimal transport and compare
physicist's and OT theory approaches, for instance the Gangbo-McCann
formula for the optimal map in terms of the Kantorovich potential is
arrived at in an intriguingly simple way by physicists
(ii) to discuss what is known rigorously about the limit problem, including
-- justification of the formal semiclassical limit [1]
-- qualitative theory of OT problems with Coulomb cost, including the
question whether ''Kantorovich minimizers'' must be ''Monge minimizers''
(yes for 2 particles, open for N particles, no for infinitely many
particles) [1,2]
-- exactly soluble cases (N=2 with radial density; N=infinity) [1, 2]
(iii) to present a natural hierarchy of further approximations of the
limit functional related to representability constraints on the pair
density which survive in the classical limit [3], and discuss the
important (open) problem of characterizing N-representable pair densities
- Supplements
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Oct 18, 2013
Friday
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09:30 AM - 10:30 AM
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Demixing in viscous fluids: a connection with optimal transportation
Felix Otto (Max-Planck-Institut für Mathematik in den Naturwissenschaften)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The demixing of a two-component fluid can be understood
as a gradient flow driven by interfacial energy and
limited by viscous dissipation. Bounds on the steepness
of the energy landscape translate into bounds on the
demixing rate. In order to understand the steepness of
the energy landscape one has to understand the distance
``in the large'' on configuration space given by the dissipation
metric ``in the small''.
It turns out that a transportation distance with logarithmic cost
is a good proxy for this distance. This observation builds on a quantitative
treatment of the DiPerna-Lions theory by DeLellis-Crippa.
- Supplements
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Otto
1.24 MB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Nonlinear inviscid damping in 2D Euler
Nader Masmoudi (New York University, Courant Institute)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- We prove the global asymptotic stability of shear flows close to planar Couette flow in the 2D incompressible Euler equations.
Specifically, given an initial perturbation of the Couette flow which is small in a suitable regularity class we show that the velocity converges strongly in L2 to another shear flow which is not far from Couette. This strong convergence is usually referred to as "inviscid
damping" and is roughly analogous to Landau damping in the Vlasov
Poisson equations. This is a joint work with Jacob Bedrossioan
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Fluids, vortex sheets, and skew-mean-curvature flows
Boris Khesin (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively.
This framework, in particular, allows one to define symplectic structures on the spaces of vortex sheets.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Relative entropy applied to shocks for Conservation Laws and applications
Alexis Vasseur (University of Texas, Austin)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- We develop a theory based on relative entropy to study the stability and contraction properties of extremal shocks of conservation laws.
- Supplements
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