PROJECTIONS ONTO THE CONE OF OPTIMAL TRANSPORT MAPS AND COMPRESSIBLE FLUID FLOWS

2010 ◽  
Vol 07 (04) ◽  
pp. 605-649 ◽  
Author(s):  
MICHAEL WESTDICKENBERG
Keyword(s):  
Euler Equations ◽  
Optimal Transport ◽  
Fluid Flows ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Polar Cone ◽  
Isentropic Euler Equations ◽  
Transport Maps ◽  
Compressible Fluid Flows

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.

scholarly journals Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

2018 ◽  
Vol 22 ◽  
pp. 35-57 ◽  
Author(s):  
Jérémie Bigot ◽  
Thierry Klein
Keyword(s):  
Optimal Transport ◽  
Probability Measures ◽  
Reference Measure ◽  
Precise Characterization ◽  
Fixed Reference ◽  
Random Probability ◽  
Wasserstein Space ◽  
Transport Maps ◽  
Random Probability Measures

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

scholarly journals All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

2011 ◽  
Vol 10 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Pierre Degond ◽  
Min Tang
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Computational Cost ◽  
Time Discretization ◽  
Implicit Time ◽  
Elliptic Type ◽  
Implicit Methods ◽  
Time Step ◽  
Low Mach Number ◽  
Isentropic Euler Equations

AbstractAn all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

scholarly journals A new variation for the relativistic Euler equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mahmoud A. E. Abdelrahman ◽  
Hanan A. Alkhidhr
Keyword(s):  
Euler Equations ◽  
Nonlinear Waves ◽  
Approximation Scheme ◽  
Strength Function ◽  
Convergent Sequence ◽  
Approximate Solutions ◽  
Large Initial Data ◽  
Initial Value ◽  
Relativistic Euler Equations ◽  
Glimm Scheme

Abstract The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

scholarly journals Ergodic theory for energetically open compressible fluid flows

2021 ◽  
Vol 423 ◽  
pp. 132914
Author(s):  
Francesco Fanelli ◽  
Eduard Feireisl ◽  
Martina Hofmanová
Keyword(s):  
Ergodic Theory ◽  
Compressible Fluid ◽  
Fluid Flows ◽  
Compressible Fluid Flows

scholarly journals Sharp boundary ε-regularity of optimal transport maps

2021 ◽  
Vol 381 ◽  
pp. 107603
Author(s):  
Tatsuya Miura ◽  
Felix Otto
Keyword(s):  
Optimal Transport ◽  
Sharp Boundary ◽  
Transport Maps

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

scholarly journals Regularity of optimal transport maps on multiple products of spheres

2013 ◽  
Vol 15 (4) ◽  
pp. 1131-1166 ◽  
Author(s):  
Alessio Figalli ◽  
Young-Heon Kim ◽  
Robert McCann
Keyword(s):  
Optimal Transport ◽  
Multiple Products ◽  
Transport Maps ◽  
Products Of Spheres
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