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Numerical enclosures of the optimal cost of the Kantorovitch’s mass transportation problem

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Abstract

The problem of optimal transportation was formalized by the French mathematician Gaspard Monge in 1781. Since Kantorovitch, this (generalized) problem is formulated with measure theory. Based on Interval Arithmetic, we propose a guaranteed discretization of the Kantorovitch’s mass transportation problem. Our discretization is spatial: supports of the two mass densities are partitioned into finite families. The problem is relaxed to a finite dimensional linear programming problem whose optimum is a lower bound to the optimum of the initial one. Based on Kantorovitch duality and Interval Arithmetic, a method to obtain an upper bound to the optimum is also provided. Preliminary results show that good approximations are obtained.

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References

  1. The GNU Multiple Precision Arithmetic Library. https://gmplib.org/

  2. GLPK: GNU Linear Programming Kit. http://www.gnu.org/software/glpk/

  3. Alefeld, G.: Inclusion methods for systems of nonlinear equations—the interval Newton method and modifications. In: Herzberger, J. (ed.) ”Topics in Validated Computation”, Proceedings of the IMACS-GAMM International Workshop on Validated Computation, Oldenburg, Germany, August 30–September 3, 1993, pp. 7–26. Elsevier, Amsterdam (1994)

  4. Benamou, J.D.: Numerical resolution of an “unbalanced” mass transport problem. ESAIM Math. Model. Numer. Anal. 37(5), 851–868 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987). French, with English summary, MR 923203 (89b:58226)

    MathSciNet  MATH  Google Scholar 

  6. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bosc, D.: Numerical approximation of optimal transport maps. SSRN (2010). doi:10.2139/ssrn.1730684

  8. Chartrand, R., Vixie, K.R., Wohlberg, B., Bollt, E.M.: A gradient descent solution to the Monge–Kantorovich problem. Appl. Math. Sci. 3(22), 1071–1080 (2009)

    MathSciNet  Google Scholar 

  9. Evans, L.C.: Partial differential equations and Monge–Kantorovich mass transfer. Current Developments in Mathematics, 1997 (Cambridge, MA), pp. 65–126. International Press, Boston (1999)

    Google Scholar 

  10. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001)

    MATH  Google Scholar 

  11. Kantorovitch, L.V.: On the translocation of masses. Doklady. Akad. Nauk SSSR 37(7–8), 227–229 (1942)

    MathSciNet  Google Scholar 

  12. Kearfott, R.B.: Interval computations: Introduction, uses, and resources. Euromath. Bull. 2(1), 95–112 (1996)

    MathSciNet  Google Scholar 

  13. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)

    Book  Google Scholar 

  14. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1996)

    Google Scholar 

  15. Moore, R.E.: Practical aspects of interval computation. Appl. Math. 13, 52–92 (1968)

    MATH  Google Scholar 

  16. Lohner, R.J.: Enclosing the solutions of ordinary initial and boundary value problems. Computer Arithmetic, pp. 255–286. Teubner, Stuttgart (1987)

    Google Scholar 

  17. Mérigot, Q.: A multiscale approach to optimal transport. Comput. Graph. Forum 30(5), 1583–1592 (2011)

    Article  Google Scholar 

  18. Neumaier, A.: Interval Methods for Systems of Equations Encyclopedia of Mathematics and its Applications, vol. 37. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  19. Revol, N.: Interval Newton iteration in multiple precision for the univariate case. Numerical Algorithms 34(2), 417–426 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Robinson, Stephen M.: Bounds for error in the solution set of a perturbed linear program. Linear Algebra and its Applications 6, 69–81 (1973)

    Article  MATH  Google Scholar 

  21. Casado, L.G., Martinez, J.A., Garcia, I., Sergeyev, Y.A.D.: New interval analysis support functions using gradient information in a global minimization algorithm. Journal of Global Optimization 25, 345–362 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci. Math. Sér. I 328, 1197–1202 (1999)

    MATH  Google Scholar 

  23. Dubuc, S., Kagabo, I.: Numerical solutions of the mass transfer problem. RAIRO Oper. Res. 40(1), 1–17 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Anderson, E.J., Philpott, A.B.: Duality and an algorithm for a class of continuous transportation problems. Math. Oper. Res. 9(2), 222–231 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gabriel, J., Escobedo-Trujillo, B.: An approximation scheme for the mass transfer problem. Morfismos 10(2), 1–13 (2006)

    Google Scholar 

  26. Gabriel, J., González-Hernández, J., López-Martínez, R.: Numerical approximations to the mass transfer problem on compact spaces. IMA J. Numer. Anal. 30, 1121–1136 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)

    Book  MATH  Google Scholar 

  28. Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2008)

    Google Scholar 

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Authors and Affiliations

  1. LARIS, 62 Avenue Notre Dame du Lac, 49000, Angers, France

    Nicolas Delanoue, Mehdi Lhommeau & Philippe Lucidarme

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  1. Nicolas Delanoue
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  2. Mehdi Lhommeau
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  3. Philippe Lucidarme
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Correspondence to Nicolas Delanoue.

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Delanoue, N., Lhommeau, M. & Lucidarme, P. Numerical enclosures of the optimal cost of the Kantorovitch’s mass transportation problem. Comput Optim Appl 63, 855–873 (2016). https://doi.org/10.1007/s10589-015-9794-9

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