Numerical resolution of an “unbalanced” mass transport problem
- Volume: 37, Issue: 5, page 851-868
- ISSN: 0764-583X
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topBenamou, Jean-David. "Numerical resolution of an “unbalanced” mass transport problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.5 (2003): 851-868. <http://eudml.org/doc/245803>.
@article{Benamou2003,
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.},
author = {Benamou, Jean-David},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Monge–Kantorovitch problem; Wasserstein distance; augmented lagrangian method; penalization technique; Monge-Kantorovitch problem; augmented Lagrangian numerical method; error estimation; meteorology},
language = {eng},
number = {5},
pages = {851-868},
publisher = {EDP-Sciences},
title = {Numerical resolution of an “unbalanced” mass transport problem},
url = {http://eudml.org/doc/245803},
volume = {37},
year = {2003},
}
TY - JOUR
AU - Benamou, Jean-David
TI - Numerical resolution of an “unbalanced” mass transport problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 5
SP - 851
EP - 868
AB - 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.
LA - eng
KW - Monge–Kantorovitch problem; Wasserstein distance; augmented lagrangian method; penalization technique; Monge-Kantorovitch problem; augmented Lagrangian numerical method; error estimation; meteorology
UR - http://eudml.org/doc/245803
ER -
References
top- [1] M. Balinski, A competitive (dual) simplex method for the assignment problem. Math. Program. 34 (1986) 125–141. Zbl0596.90064
- [2] F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998) 335–361. Zbl0901.26010
- [3] J.-D. Benamou, A domain decomposition method for the polar factorization of vector valued mappings. SIAM J. Numer. Anal. 32 (1995) 1808–1838. Zbl0852.65012
- [4] 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).
- [5] J.D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58 (1998) 1450–1461. Zbl0915.35024
- [6] J.D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. Zbl0968.76069
- [7] J.D. Benamou and Y. Brenier, Mixed /Wasserstein Optimal Mapping Between Prescribed Densities Functions (submitted). Zbl1010.49029
- [8] J.D. Benamou, Y. Brenier and K. Guittet, Numerical resolution of a multiphasic optimal mass transport problem. Tech. Report INRIA RR-4022. Zbl1135.49025
- [9] G. Boucjitte, G. Buttazzo and P. Seppechere, Shape Optimization Solutions via Monge-Kantorovich. C. R. Acad. Sci. Paris Sér. I 324 (1997) 1185–1191. Zbl0884.49023
- [10] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991) 375–417. Zbl0738.46011
- [11] Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52 (1999) 411–452. Zbl0910.35098
- [12] Y. Brenier, Extended Monge-Kantorovich theory. CIME 2001 lecture. Zbl1064.49036
- [13] L.A. Caffarelli, Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45 (1992) 1141–1151. Zbl0778.35015
- [14] L.A. Caffarelli, Boundary regularity of maps with convex potentials. II. Ann. of Math. 144 (1996) 3, 453–496. Zbl0916.35016
- [15] M.J.P. Cullen, Solution to a model of a front forced by deformation. Q. J. R. Met. Soc. 109 (1983) 565–573.
- [16] M.J.P. Cullen, private communication.
- [17] M.J.P. Cullen and R.J. Purser, An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmopheric Sci. 41 (1984) 1477–1497.
- [18] R.J. Douglas, Decomposition of weather forecast error using rearrangements of functions. (Preprint.)
- [19] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer. Lecture notes. Zbl0954.35011
- [20] 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. Zbl0525.65045MR724072
- [21] U. Frisch et al., Back to the early Universe by optimal mass transportation. Nature 417 (2002) 260–262.
- [22] W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. Zbl0887.49017
- [23] W. Gangbo and R.J. McCann, Shape recognition via Wasserstein distance. Quart. Appl. Math. 58 (2000) 705–737. Zbl1039.49038
- [24] K. Guittet, On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques. SIAM J. Numer. Anal. 41 (2003) 382–399. Zbl1039.65050
- [25] K. Guittet, Ph.D. dissertation (2002).
- [26] S. Haker, A. Tannenbaum and R. Kikinis, Mass preserving mapping and image registration. MICCAI (2001) 120–127. Zbl1041.68621
- [27] R. Jonker and A. Volgenant, A shortest augmenting path algorithm for dense and sparse linear assignment problem. Computing 38 (1987) 325–340. Zbl0607.90056
- [28] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. Zbl0915.35120
- [29] T. Kaijser, Computing the Kantorovich distance for images. J. Math. Imaging Vision 9 (1998) 173–198. Zbl0911.68207
- [30] L.V. Kantorovich, On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942) 199–201. Zbl0061.09705
- [31] D. Kinderlehrer and N. Walkington, Approximation of Parabolic Equations based upon a Wasserstein metric. ESAIM: M2AN 33 (1999) 837–852. Zbl0936.65121
- [32] S.A. Kochengin and V.I. Oliker, Determination of reflector surfaces from near-field scattering data. Inverse Problems 13 (1997) 363–373. Zbl0871.35107
- [33] R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589–608. Zbl1011.58009
- [34] R. Menozzi, Utilisation de la distance de Wasserstein et application sismique. Rapport IUP Génie Mathématique et Informatique, Université Paris IX-Dauphine.
- [35] G. Monge, Mémoire sur la théorie des déblais et des remblais. Mem. Acad. Sci. Paris (1781).
- [36] F. Otto, The geometry of dissipative evolution equation: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101–174. Zbl0984.35089
- [37] S.T. Rachev and L. Rüschendorf, Mass transportation problems, in Theory, Probability and its Applications, Vol. I. Springer-Verlag, New York (1998) 508. Zbl0990.60500MR1619170
- [38] A. Shnirelman, Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4 (1994) 586–620. Zbl0851.76003
- [39] C. Villani, Topics in mass transport. Lecture notes (2000).
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