Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T23:05:05.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

Part of: Numerical methods in calculus of variations and optimal control Numerical analysis in abstract spaces Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 September 2018

JEAN-DAVID BENAMOU  and
VINCENT DUVAL
JEAN-DAVID BENAMOU
Affiliation:
INRIA Paris (MOKAPLAN) and CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France emails: jean-david.benamou@inria.fr; vincent.duval@inria.fr
VINCENT DUVAL
Affiliation:
INRIA Paris (MOKAPLAN) and CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France emails: jean-david.benamou@inria.fr; vincent.duval@inria.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

Keywords

Optimal transportMonge-Ampère equationfinite-difference scheme

MSC classification

Primary: 65N06: Finite difference methods
Secondary: 65J15: Equations with nonlinear operators (do not use ) 49M15: Newton-type methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1041 - 1078
Copyright
© Cambridge University Press 2018 

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

Barles, G. & Souganidis, P. E. (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4(3), 271–283.Google Scholar
Benamou, J.-D. & Brenier, Y. (2000) A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393.CrossRefGoogle Scholar
Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L. & Peyré, G. (2015) Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comp. 2(37), A1111–A1138.CrossRefGoogle Scholar
Benamou, J.-D., Collino, F. & Mirebeau, J.-M. (2016) Monotone and consistent discretization of the Monge–Ampere operator. Math. Comput. 85(302), 2743–2775.CrossRefGoogle Scholar
Benamou, J.-D., Froese, B. D. & Oberman, A. M. (2014) Numerical solution of the optimal transportation problem using the Monge–Ampère equation. J. Comput. Phys. 260, 107–126.CrossRefGoogle Scholar
Brenier, Y. (1991) Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417.CrossRefGoogle Scholar
Caffarelli, L. A. (1992) Boundary regularity of maps with convex potentials. Commun. Pure Appl. Math. 45(9), 1141–1151.CrossRefGoogle Scholar
Caffarelli, L. A. (1996) Boundary regularity of maps with convex potentials – II. Ann. Math. 144(3), 453–496.CrossRefGoogle Scholar
Carlier, G., Mérigot, Q., Oudet, E. & Benamou, J.-D. (2016) Discretization of functionals involving the Monge-Ampère operator. Numer. Math. 134(3), 611–636.Google Scholar
Chizat, L., Peyré, G., Schmitzer, B. & Vialard, F.-X. (2016) Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.Google Scholar
Crandall, M. G., Ishii, H. & Lions, P.-L. (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67.CrossRefGoogle Scholar
Cullen, M. J. P. & Purser, R. J. (1984) An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmos. Sci. 41(9), 1477–1497.2.0.CO;2>CrossRefGoogle Scholar
Cuturi, M. (2013) Sinkhorn distances: lightspeed computation of optimal transport. In: Burges, C. J. C., Bottou, L., Welling, M., Ghahramani, Z. & Weinberger, K. Q. (editors), Proceedings of NIPS, Curran Associates, Inc., pp. 2292–2300.Google Scholar
De Philippis, G. & Figalli, A. (2014) Partial Regularity Results in Optimal Transportation. In: Ancona, V., Strickland, E. (editors), Trends in Contemporary Math., Springer INdAM Series, Springer, 8. https://doi.org/10.1007/978-3-319-05254-0_21.Google Scholar
Figalli, A. (2010) Regularity properties of optimal maps between nonconvex domains in the plane. Commun. Partial Differ. Equations 35(3), 465–479.CrossRefGoogle Scholar
Figalli, A. & Loeper, G. (2009) C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two. Calculus Var. Partial Differ. Equations 35, 537–550.CrossRefGoogle Scholar
Froese, B. D. & Oberman, A. M. (2013) Convergent filtered schemes for the Monge–Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423–444.CrossRefGoogle Scholar
Galichon, A. (2016) Optimal transport methods in economics, Princeton University Press.CrossRefGoogle Scholar
Gutierez, C. E. (2001) The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Basel.Google Scholar
Hiriart-Urruty, J.-B. & Lemaréchal, C. (1996) Convex Analysis and Minimization Algorithms, Springer Verlag, Heidelberg. Two volumes, 2nd printing.Google Scholar
Kantorovich, L. (1942) On the transfer of masses. Dokl. Acad. Nauk. USSR 37, 7–8.Google Scholar
Kitagawa, J., Mérigot, Q. & Thibert, B. (2016) Convergence of a Newton algorithm for semi-discrete optimal transport. arXiv preprint arXiv:1603.05579.Google Scholar
Knott, M. & Smith, C. S. (1984) J. Optim. Theory Appl. 43, 39.Google Scholar
Lévy, B. (2015) A numerical algorithm for l2 semi-discrete optimal transport in 3D. ESAIM: Math. Modell. Numer. Anal. 49(6), 1693–1715.CrossRefGoogle Scholar
LeVeque, R. J. (1992) Numerical Methods for Conservation Laws, 2nd ed. Lectures in Mathematics, Birkhäuser.CrossRefGoogle Scholar
Loeper, G. & Rapetti, F. (2005) Numerical Solution of the Monge-Ampère Equation by a Newton’s Algorithm. Comptes Rendus Mathématique, Académie des Sciences, Paris.CrossRefGoogle Scholar
McCann, R. J. (1997) A convexity principle for interacting gases. Adv. Math. 128(1), 153–179.CrossRefGoogle Scholar
Mérigot, Q. (2011) A multiscale approach to optimal transport. Comput. Graph. Forum 30(5), 1583–1592.CrossRefGoogle Scholar
Mirebeau, J.-M. (2015) Discretization of the 3D Monge-Ampere operator, between wide stencils and power diagrams. ESAIM Math. Model. Numer. Anal. 49(5), 1511–1523.CrossRefGoogle Scholar
Mirebeau, J.-M. (2016) Adaptive, anisotropic and hierarchical cones of discrete convex functions. Numer. Math. 132(4), 807–853.CrossRefGoogle Scholar
Oberman, A. M. (2006) Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895.CrossRefGoogle Scholar
Oberman, A. M. (2008) Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18(5), 759–780.CrossRefGoogle Scholar
Oberman, A. M. & Ruan, Y. (2015) An efficient linear programming method for optimal transportation. arXiv: 1509.03668.Google Scholar
Oliker, V. I. & Prussner, L. D. (1988) On the numerical solution of the equation (∂ 2z/∂x 2)(∂ 2z/∂y 2) − ((∂ 2z/∂x∂y))2 = f and its discretizations. I. Numer. Math. 54(3), 271–293.CrossRefGoogle Scholar
Rockafellar, R. T. (1996) Convex Analysis, Vol. 28, Princeton University Press.Google Scholar
Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians, Birkhäuser, New York.CrossRefGoogle Scholar
Saumier, L.-P., Agueh, M. & Khouider, B. (2015) An efficient numerical algorithm for the l2 optimal transport problem with periodic densities. IMA J. Appl. Math. 80(1), 135–157.CrossRefGoogle Scholar
Schneider, R. (2013) Convex Bodies: The Brunn-Minkowski Theory. 2nd ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
Schmitzer, B. (2016) A sparse multiscale algorithm for dense optimal transport. J. Math. Imaging Vis. 56(2), 238–259.CrossRefGoogle Scholar
Urbas, J. I. E. (1990) On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations. Indiana Univ. Math. J. 39(2), 355–382.CrossRefGoogle Scholar
Villani, C. (2003) Topics in Optimal Transportation. Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI.Google Scholar
Villani, C. (2009) Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin.CrossRefGoogle Scholar