[CvGmt News] [Fwd: Post-Doc Proposal at MOKAPLAN (INRIA-PARIS DAUPHINE)]

[buttazzo at dm.unipi.it]

-------------------------- Messaggio originale ---------------------------
Oggetto: Post-Doc Proposal at MOKAPLAN (INRIA-PARIS DAUPHINE)
Da:      "jdb" <jean-david.benamou at inria.fr>
Data:    Ven, 22 Marzo 2013 16:32
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Dear Colleagues,
please find below a proposal for a one-year postdoctoral position in 
Paris for the next academic year.
Do not hesitate to circulate.
Thank you and best regards,
Jean-David Benamou and Guillaume Carlier
zuazua at bcamath.org

This is a proposal for a one year Post-Doctoral position at
INRIA-Paris/Rocquencourt in the MOKAPLAN team and
under the supervision of G. Carlier and J.-D. Benamou.

The starting date is flexible. May be as early as sept. 2013.  Net Salary
is  2137euros per month.

The post-Doc is funded by the  French ANR  ISOTACE consortium which also 
involves U. Orsay (B. Maury, F. Santambroggio)
 U. Paris 7 (Y. Achdou), Ecole Polytechnique (N. Touzi, Y. Brenier) and  2
small companies.

Scientific description :

The post doc  objective is to design and implement robust and efficient
numerical methods for a large class of  variational problems
including the Gradient Flow formulation of Non linear diffusion PDEs
(Granular Flows
Keller Segel equation ...), new modelizations of Economic problems
(Cournot-Nash Principal-Agent, Urban design ...).

These problems optimize a density which is mapped or matched to some
compatible data.
The density displacement   arises from the gradient of a potential
constrained to be convex. The convexity constraint  makes the connection
with optimal transportation theory.

The discretization and implementation of the convexity constraint is a
challenging problem
wich has recently received some attention (Ekeland-Moreno
Carlier-Maury-Lachand Robert  Oberman ... )
The candidate will first study/implement/compare existing methods. We will
then
try to select or invent  the most efficient approach for our target problem.

The central task is to address numerically the problem of minimizing a
displacement convex functional  (in the sense of McCann)  with a
L2-Wasserstein penalization term.  Taking advantage of the properties of
the variational problems
 in the space of displacement, the candidate will study/implement
gradient descent methods where the optimization variable is
the convexity constrained displacement potential or gradient.

This position is strongly research oriented with plenty of opportunities
to explore connected problems.  The candidate will preferably have
a previous experience on optimal transportation or infinite dimensional
optimization.  Scientific programming experience is necessary.

Bibliography

- P. Chon� and Rochet J.-C. Ironing, sweeping and multidimensional
screening. Econometrica, 1998.
- G. Carlier, T. Lachand-Robert: Regularity of solutions for some
variational problems subject to a convexity constraint, Comm. Pure Appl.
Math. 54 (2001) 583�594.
- Carlier, G.; Lachand-Robert, T.; Maury, B. A numerical approach to
variational problems subject to convexity constraint. Numer. Math. 88
(2001), no. 2, 299�318.
- Ekeland, Ivar; Moreno-Bromberg, Santiago An algorithm for computing
solutions of variational problems with global convexity constraints.
Numer. Math. 115 (2010), no. 1, 45�69.
- Oberman, Adam; A numerical method for variational problems with
convexity constraints  http://arxiv.org/abs/1107.5290v3
- Merigot Quentin, Oudet E. ; private communication.
- Mirebeau Jean-Marie ; private communication.
- Blanchet A. Carlier G. ; Optimal transport and Cournot-Nash Equilibria
http://www.ceremade.dauphine.fr/~carlier/blanchetcarlierfinal.pdf  .
- Aguilera, N�stor E.; Morin, Pedro On convex functions and the finite
element method. SIAM J. Numer. Anal. 47 (2009), no. 4, 3139�3157.
- Aguilera, N�stor E.; Morin, Pedro Approximating optimization problems
over convex functions. Numer. Math. 111 (2008), no. 1, 1�34.
- Jordan, Richard; Kinderlehrer, David; Otto, Felix The variational
formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998),
no. 1, 1�17.
- Kinderlehrer, David; Walkington, Noel J. Approximation of parabolic
equations using the Wasserstein metric. M2AN Math. Model. Numer. Anal. 33
(1999), no. 4, 837�852.
- J. A. Carrillo, J. S. Moll, Numerical simulation of diffusive and
aggregation phenomena in nonlinear continuity equations by evolving
diffeomorphisms, SIAM J. Sci. Comput. 31, 4305-4329, 2009.
- A. Blanchet, E. A. Carlen, J. A. Carrillo, Functional inequalities,
thick tails and asymptotics for the critical mass Patlak-Keller-Segel
model, J. Func. Anal. 262, 2142-2230, 2012.
...