Mersenne banner

Livres, Actes et Séminaires du Centre Mersenne

  • Livres
  • Séminaires
  • Congrès
  • Tout
  • Auteur
  • Titre
  • Bibliographie
  • Plein texte
NOT
Entre et
  • Tout
  • Auteur
  • Titre
  • Date
  • Bibliographie
  • Mots-clés
  • Plein texte
  • Précédent
  • Journées équations aux dérivées partielles
  • Année 2015
  • article no. 7
  • Suivant
Numerical resolution of Euler equations through semi-discrete optimal transport
Jean-Marie Mirebeau1
1 CNRS et Université Paris-Sud Université Paris-Saclay Départment de Mathématiques Bâtiment 425 91405 Orsay Cedex France
Journées équations aux dérivées partielles (2015), article no. 7, 16 p.
  • Résumé

Geodesics along the group of volume preserving diffeomorphisms are solutions to Euler equations of inviscid incompressible fluids, as observed by Arnold [4]. On the other hand, the projection onto volume preserving maps amounts to an optimal transport problem, as follows from the generalized polar decomposition of Brenier [14].

We present, in the first section, the framework of semi-discrete optimal transport, initially developed for the study of generalized solutions to optimal transport [1] and now regarded as an efficient approach to computational optimal transport. In a second and largely independent section, we present numerical approaches for Euler equations seen as a boundary value problem [16, 7, 33]: knowing the initial and final positions of some fluid particles, reconstruct intermediate fluid states. Depending on the data, we either recover a classical solution to Euler equations, or a generalized flow [15] for which the fluid particles motion is non-deterministic, as predicted by [39].

  • Détail
  • Export
  • Comment citer
DOI : 10.5802/jedp.636
Affiliations des auteurs :
Jean-Marie Mirebeau 1

1 CNRS et Université Paris-Sud Université Paris-Saclay Départment de Mathématiques Bâtiment 425 91405 Orsay Cedex France
  • BibTeX
  • RIS
  • EndNote
@incollection{JEDP_2015____A7_0,
     author = {Jean-Marie Mirebeau},
     title = {Numerical resolution of {Euler} equations  through semi-discrete optimal transport},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {7},
     pages = {1--16},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2015},
     doi = {10.5802/jedp.636},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.636/}
}
TY  - JOUR
AU  - Jean-Marie Mirebeau
TI  - Numerical resolution of Euler equations  through semi-discrete optimal transport
JO  - Journées équations aux dérivées partielles
PY  - 2015
SP  - 1
EP  - 16
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.636/
DO  - 10.5802/jedp.636
LA  - en
ID  - JEDP_2015____A7_0
ER  - 
%0 Journal Article
%A Jean-Marie Mirebeau
%T Numerical resolution of Euler equations  through semi-discrete optimal transport
%J Journées équations aux dérivées partielles
%D 2015
%P 1-16
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.636/
%R 10.5802/jedp.636
%G en
%F JEDP_2015____A7_0
Jean-Marie Mirebeau. Numerical resolution of Euler equations  through semi-discrete optimal transport. Journées équations aux dérivées partielles (2015), article  no. 7, 16 p. doi : 10.5802/jedp.636. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.636/
  • Bibliographie
  • Cité par

[1] R Abgrall Construction of Simple, Stable, and Convergent High Order Schemes for Steady First Order Hamilton–Jacobi Equations, SIAM Journal on Scientific Computing, Volume 31 (2009) no. 4, pp. 2419-2446 | MR | Zbl

[2] N E Aguilera; P Morin On Convex Functions and the Finite Element Method, SIAM Journal on Numerical Analysis, Volume 47 (2009) no. 4, pp. 3139-3157 | MR | Zbl

[3] L Ambrosio; A Figalli On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations, Calculus of Variations and Partial Differential Equations, Volume 31 (2007) no. 4, pp. 497-509 | MR | Zbl

[4] V Arnold Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’institut Fourier, Volume 16 (1966) no. 1, pp. 319-361 | Numdam | MR | Zbl

[5] F Aurenhammer; F Hoffmann; B Aronov Minkowski-Type Theorems and Least-Squares Clustering, Algorithmica, Volume 20 (1998) no. 1, pp. 61-76 | MR | Zbl

[6] J D Benamou; Y Brenier; K Guittet The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation, International Journal for Numerical Methods in Fluids, Volume 40 (2002) no. 1-2, pp. 21-30 | MR | Zbl

[7] J D Benamou; G Carlier; M Cuturi; G Peyré; L Nenna Iterative Bregman projections for regularized transportation problems, Sci. Comp, 2015 | MR

[8] J-D Benamou; F Collino; J-M Mirebeau Monotone and Consistent discretization of the Monge-Ampere operator, Mathematics of computation (2015)

[9] J-D Benamou; B D Froese; A M Oberman Two numerical methods for the elliptic Monge-Ampere equation, M2AN. Mathematical Modelling and Numerical Analysis, Volume 44 (2010) no. 4, pp. 737-758 | Numdam | MR | Zbl

[10] J-D Benamou; B D Froese; A M Oberman Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation, Journal of Computational Physics, Volume 260 (2014), pp. 107-126 | MR

[11] M Bernot; A Figalli; F Santambrogio Generalized solutions for the Euler equations in one and two dimensions, Journal de Mathématiques Pures et Appliquées, Volume 91 (2009) no. 2, pp. 137-155 | MR | Zbl

[12] Y Brenier A combinatorial algorithm for the Euler equations of incompressible flows, Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987) (1989), pp. 325-332 | MR | Zbl

[13] Y Brenier The least action principle and the related concept of generalized flows for incompressible perfect fluids, Journal of the American Mathematical Society, Volume 2 (1989) no. 2, pp. 225-255 | MR | Zbl

[14] Y Brenier Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, Volume 44 (1991) no. 4, pp. 375-417 | MR | Zbl

[15] Y Brenier The dual least action principle for an ideal, incompressible fluid , Archive for rational mechanics and analysis (1993)

[16] Y Brenier Generalized solutions and hydrostatic approximation of the Euler equations, Physica D: Nonlinear Phenomena (2008) | MR | Zbl

[17] M Bruveris; François X Vialard On Completeness of Groups of Diffeomorphisms (2014) (http://arxiv.org/abs/1403.2089)

[18] A Carverhill; F J Pedit Global solutions of the Navier-Stokes equation with strong viscosity, Annals of Global Analysis and Geometry, Volume 10 (1992) no. 3, pp. 255-261 | MR | Zbl

[19] CGAL (http://www.cgal.org/)

[20] P M M De Castro; Q Merigot; B Thibert Intersection of paraboloids and application to Minkowski-type problems, Computational geometry (SoCG’14), ACM, New York, 2014, pp. 308-317 | MR

[21] F de Goes; K Breeden; V Ostromoukhov; M Desbrun Blue noise through optimal transport, ACM Transactions on Graphics (TOG), Volume 31 (2012) no. 6, pp. 171

[22] F de Goes; C Wallez; J Huang; U Zhejiang; D Pavlov Power Particles: An incompressible fluid solver based on power diagrams (2015) (geometry.caltech.edu)

[23] C De Lellis; L Székelyhidi Jr The Euler equations as a differential inclusion, Annals of mathematics (2009)

[24] L Euler Theoria Motus Corporum Solidorum Seu Rigidorum (1765)

[25] A Figalli; S Daneri Variational models for the incompressible Euler equations, HCDTE Lecture Notes, Part II (2012) | MR

[26] B D Froese; A M Oberman Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher, SIAM Journal on Numerical Analysis, Volume 49 (2011) no. 4, pp. 1692-1714 | MR | Zbl

[27] L V Kantorovich Kantorovich: On the transfer of masses, Dokl. Akad. Nauk. SSSR, 1942

[28] H-J Kuo; N S Trudinger Discrete Methods for Fully Nonlinear Elliptic Equations, SIAM Journal on Numerical Analysis, Volume 29 (1992) no. 1, pp. 123-135 | MR | Zbl

[29] B Lévy A numerical algorithm for L 2 semi-discrete optimal transport in 3D (2014) (http://arxiv.org/abs/1409.1279v1)

[30] G Loeper; F Rapetti Numerical solution of the Monge-Ampère equation by a Newton’s algorithm, Comptes Rendus Mathématique. Académie des Sciences. Paris, Volume 340 (2005) no. 4, pp. 319-324 | MR | Zbl

[31] X-N Ma; N S Trudinger; X-J Wang Regularity of Potential Functions of the Optimal Transportation Problem, Archive for rational mechanics and analysis, Volume 177 (2005) no. 2, pp. 151-183 | MR | Zbl

[32] Q Merigot A Multiscale Approach to Optimal Transport, Computer Graphics Forum, Volume 30 (2011) no. 5, pp. 1583-1592

[33] Q Merigot; J-M Mirebeau Minimal geodesics along volume preserving maps, through semi-discrete optimal transport (2015) (http://arxiv.org/abs/1505.03306)

[34] Q Merigot; E Oudet Handling convexity-like constraints in variational problems, SIAM Journal on Numerical Analysis, Volume 52 (2014) no. 5, pp. 2466-2487 | MR

[35] J-M Mirebeau Adaptive, anisotropic and hierarchical cones of discrete convex functions, Numerische Mathematik (2015), pp. 1-47

[36] A M Oberman; Y Ruan An efficient linear programming method for Optimal Transportation (2015) (http://arxiv.org/abs/1509.03668)

[37] V I Oliker; L D Prussner On the numerical solution of the equation ∂ 2 z ∂x 2 ∂ 2 z ∂y 2 -∂ 2 z ∂x∂y 2 =f and its discretizations, I, Numerische Mathematik, Volume 54 (1989) no. 3, pp. 271-293 | MR | Zbl

[38] B Schmitzer A sparse algorithm for dense optimal transport, Scale Space and Variational Methods in Computer Vision, Springer International Publishing, Cham, 2015, pp. 629-641

[39] A I Shnirelman Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, Volume 4 (1994) no. 5, pp. 586-620 | MR | Zbl

[40] C Villani Optimal transport: Old and new, Springer-Verlag, Berlin, 2009, pp. xxii+973 | DOI | MR | Zbl

Cité par Sources :

Diffusé par : Publié par : Développé par :
  • Nous suivre