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].
@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/
[1] 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] On Convex Functions and the Finite Element Method, SIAM Journal on Numerical Analysis, Volume 47 (2009) no. 4, pp. 3139-3157 | MR | Zbl
[3] 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] 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] Minkowski-Type Theorems and Least-Squares Clustering, Algorithmica, Volume 20 (1998) no. 1, pp. 61-76 | MR | Zbl
[6] 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] Iterative Bregman projections for regularized transportation problems, Sci. Comp, 2015 | MR
[8] Monotone and Consistent discretization of the Monge-Ampere operator, Mathematics of computation (2015)
[9] 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] Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation, Journal of Computational Physics, Volume 260 (2014), pp. 107-126 | MR
[11] 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] 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] 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] 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] The dual least action principle for an ideal, incompressible fluid , Archive for rational mechanics and analysis (1993)
[16] Generalized solutions and hydrostatic approximation of the Euler equations, Physica D: Nonlinear Phenomena (2008) | MR | Zbl
[17] On Completeness of Groups of Diffeomorphisms (2014) (http://arxiv.org/abs/1403.2089)
[18] 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] (http://www.cgal.org/)
[20] Intersection of paraboloids and application to Minkowski-type problems, Computational geometry (SoCG’14), ACM, New York, 2014, pp. 308-317 | MR
[21] Blue noise through optimal transport, ACM Transactions on Graphics (TOG), Volume 31 (2012) no. 6, pp. 171
[22] Power Particles: An incompressible fluid solver based on power diagrams (2015) (geometry.caltech.edu)
[23] The Euler equations as a differential inclusion, Annals of mathematics (2009)
[24] Theoria Motus Corporum Solidorum Seu Rigidorum (1765)
[25] Variational models for the incompressible Euler equations, HCDTE Lecture Notes, Part II (2012) | MR
[26] 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] Kantorovich: On the transfer of masses, Dokl. Akad. Nauk. SSSR, 1942
[28] Discrete Methods for Fully Nonlinear Elliptic Equations, SIAM Journal on Numerical Analysis, Volume 29 (1992) no. 1, pp. 123-135 | MR | Zbl
[29] A numerical algorithm for semi-discrete optimal transport in 3D (2014) (http://arxiv.org/abs/1409.1279v1)
[30] 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] 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] A Multiscale Approach to Optimal Transport, Computer Graphics Forum, Volume 30 (2011) no. 5, pp. 1583-1592
[33] Minimal geodesics along volume preserving maps, through semi-discrete optimal transport (2015) (http://arxiv.org/abs/1505.03306)
[34] Handling convexity-like constraints in variational problems, SIAM Journal on Numerical Analysis, Volume 52 (2014) no. 5, pp. 2466-2487 | MR
[35] Adaptive, anisotropic and hierarchical cones of discrete convex functions, Numerische Mathematik (2015), pp. 1-47
[36] An efficient linear programming method for Optimal Transportation (2015) (http://arxiv.org/abs/1509.03668)
[37] On the numerical solution of the equation and its discretizations, I, Numerische Mathematik, Volume 54 (1989) no. 3, pp. 271-293 | MR | Zbl
[38] A sparse algorithm for dense optimal transport, Scale Space and Variational Methods in Computer Vision, Springer International Publishing, Cham, 2015, pp. 629-641
[39] Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, Volume 4 (1994) no. 5, pp. 586-620 | MR | Zbl
[40] Optimal transport: Old and new, Springer-Verlag, Berlin, 2009, pp. xxii+973 | DOI | MR | Zbl
Cité par Sources :