Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-06T22:07:40.758Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical resolution of an “unbalanced” mass transport problem

Published online by Cambridge University Press:  15 November 2003

Jean-David Benamou
Jean-David Benamou*
Affiliation:
INRIA-Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. jean-david.benamou@inria.fr.
Get access

Abstract

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Keywords

Monge–Kantorovitch problemWasserstein distanceaugmented Lagrangian method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 5 , September 2003 , pp. 851 - 868
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balinski, M., A competitive (dual) simplex method for the assignment problem. Math. Program. 34 (1986) 125-141. CrossRef
Barthe, F., On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998) 335-361. CrossRef
Benamou, J.-D., A domain decomposition method for the polar factorization of vector valued mappings. SIAM J. Numer. Anal. 32 (1995) 1808-1838. CrossRef
J.D. Benamou and Y. Brenier, Numerical resolution on a massively parallel computer of a test problem in meteorology using a domain decomposition algorithm, in First European conference in computational fluid dynamics. North Holland (1992).
Benamou, J.D. and Brenier, Y., Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58 (1998) 1450-1461. CrossRef
Benamou, J.D. and Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. CrossRef
J.D. Benamou and Y. Brenier, Mixed L 2 /Wasserstein Optimal Mapping Between Prescribed Densities Functions (submitted).
J.D. Benamou, Y. Brenier and K. Guittet, Numerical resolution of a multiphasic optimal mass transport problem. Tech. Report INRIA RR-4022.
Boucjitte, G., Buttazzo, G. and Seppechere, P., Shape Optimization Solutions via Monge-Kantorovich. C. R. Acad. Sci. Paris Sér. I 324 (1997) 1185-1191. CrossRef
Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991) 375-417. CrossRef
Brenier, Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52 (1999) 411-452. 3.0.CO;2-3>CrossRef
Y. Brenier, Extended Monge-Kantorovich theory. CIME 2001 lecture.
Caffarelli, L.A., Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45 (1992) 1141-1151. CrossRef
L.A. Caffarelli, Boundary regularity of maps with convex potentials. II. Ann. of Math. 144 (1996) 3, 453-496.
Cullen, M.J.P., Solution to a model of a front forced by deformation. Q. J. R. Met. Soc. 109 (1983) 565-573. CrossRef
M.J.P. Cullen, private communication.
Cullen, M.J.P. and Purser, R.J., An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmopheric Sci. 41 (1984) 1477-1497. 2.0.CO;2>CrossRef
R.J. Douglas, Decomposition of weather forecast error using rearrangements of functions. (Preprint.)
L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer. Lecture notes.
M. Fortin and R. Glowinski, Augmented Lagrangian methods. Applications to the numerical solution of boundary value problems. North-Holland Publishing Co. Studies in Mathematics and its Applications 15 (1983) 340.
Frisch, U. et al., Back to the early Universe by optimal mass transportation. Nature 417 (2002) 260-262. CrossRef
Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. CrossRef
Gangbo, W. and McCann, R.J., Shape recognition via Wasserstein distance. Quart. Appl. Math. 58 (2000) 705-737. CrossRef
Guittet, K., On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques. SIAM J. Numer. Anal. 41 (2003) 382-399. CrossRef
K. Guittet, Ph.D. dissertation (2002).
S. Haker, A. Tannenbaum and R. Kikinis, Mass preserving mapping and image registration. MICCAI (2001) 120-127.
Jonker, R. and Volgenant, A., A shortest augmenting path algorithm for dense and sparse linear assignment problem. Computing 38 (1987) 325-340. CrossRef
Jordan, R., Kinderlehrer, D. and Otto, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17. CrossRef
Kaijser, T., Computing the Kantorovich distance for images. J. Math. Imaging Vision 9 (1998) 173-198. CrossRef
Kantorovich, L.V., On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942) 199-201.
Kinderlehrer, D. and Walkington, N., Approximation of Parabolic Equations based upon a Wasserstein metric. ESAIM: M2AN 33 (1999) 837-852. CrossRef
Kochengin, S.A. and Oliker, V.I., Determination of reflector surfaces from near-field scattering data. Inverse Problems 13 (1997) 363-373. CrossRef
McCann, R.J., Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. CrossRef
R. Menozzi, Utilisation de la distance de Wasserstein et application sismique. Rapport IUP Génie Mathématique et Informatique, Université Paris IX-Dauphine.
G. Monge, Mémoire sur la théorie des déblais et des remblais. Mem. Acad. Sci. Paris (1781).
Otto, F., The geometry of dissipative evolution equation: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174. CrossRef
S.T. Rachev and L. Rüschendorf, Mass transportation problems, in Theory, Probability and its Applications, Vol. I. Springer-Verlag, New York (1998) 508.
Shnirelman, A., Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4 (1994) 586-620. CrossRef
C. Villani, Topics in mass transport. Lecture notes (2000).