Corpus ID: 85543500

Limits of density-constrained optimal transport

@article{Gladbach2019LimitsOD,
  title={Limits of density-constrained optimal transport},
  author={Peter Gladbach and Eva Kopfer},
  journal={arXiv: Analysis of PDEs},
  year={2019}
}
We consider the problem of dynamic optimal transport with a density constraint. We derive variational limits in terms of $\Gamma$-convergence for two singular phenomena. First, for densities constrained near a hyperplane we recover the optimal flow through an infinitesimal permeable membrane. Second, for rapidly oscillating periodic constraints we obtain the optimal flow through a homogenized porous medium. 
1 Citations

Figures from this paper

Homogenisation of dynamical optimal transport on periodic graphs
TLDR
A homogenisation result is derived from a Γ-convergence result for action functionals on curves of measures that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. Expand

References

SHOWING 1-10 OF 29 REFERENCES
A-QUASICONVEXITY: RELAXATION AND HOMOGENIZATION
Integral representation of relaxed energies and of Γ -limits of functionals are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first orderExpand
THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we showExpand
Transport Equations and Multi-D Hyperbolic Conservation Laws
I.- Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields.- II.- A Note on Alberti's Rank-One Theorem.- III.- RegularizingExpand
First Order Mean Field Games with Density Constraints: Pressure Equals Price
TLDR
A minimal regularity is obtained, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for the authors' model. Expand
Euler sprays and Wasserstein geometry of the space of shapes
We study a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesicExpand
Optimal transportation with capacity constraints
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, whereExpand
Stochastic Differential Equations
We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusionExpand
Least action principles for incompressible flows and optimal transport between shapes
As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in anExpand
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
Summary. The $L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.
Variational Mean Field Games
This paper is a brief presentation of those mean field games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochasticExpand
...
1
2
3
...