Advertisement

Optimal Mass Transport over Bridges

  • Yongxin Chen
  • Tryphon Georgiou
  • Michele PavonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

Abstract

We present an overview of our recent work on implementable solutions to the Schrödinger bridge problem and their potential application to optimal transport and various generalizations.

Keywords

Optimal Transport Stochastic Oscillator Optimal Transport Problem Bridge Problem Optimal Mass Transport 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)zbMATHGoogle Scholar
  2. 2.
    Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Num. Math. 84(3), 375–393 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blaquière, A.: Controllability of a Fokker-Planck equation, the Schrödinger system, and a related stochastic optimal control. Dyn. Control 2(3), 235–253 (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Georgiou, T.T., Pavon, M.: Positive contraction mappings for classical and quantum Schrödinger systems, May 2014, arXiv:1405.6650v2. J. Math. Phys. to appear
  6. 6.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution, Part I, arXiv:1408.2222v1. IEEE Trans. Aut. Control, to appear
  7. 7.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of inertial particles diffusing anisotropically with losses, arXiv:1410.1605v1, ACC Conf. (2015)
  8. 8.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic systemto a final probability distribution, Part II, arXiv:1410.3447v1. IEEE Trans. Aut. Control, to appear
  9. 9.
    Chen, Y., Georgiou, T.T., Pavon, M.: Fast cooling for a system of stochastic oscillators, Nov 2014, arXiv:1411.1323v1
  10. 10.
    Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint, arXiv:1412.4430v1
  11. 11.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system, arXiv:1502.01265v1
  12. 12.
    Chen, Y., Georgiou, T.T., Pavon, M.: A computational approach to optimal mass transport via the Schrödinger bridge problem (2015, in preparation)Google Scholar
  13. 13.
    Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dai Pra, P., Pavon, M.: On the Markov processes of Schroedinger, the Feynman-Kac formula and stochastic control. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds.) Realization and Modeling in System Theory - Proceedings of the 1989 MTNS Conference, pp. 497–504. Birkaeuser, Boston (1990)Google Scholar
  15. 15.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Applied Math., vol. 38, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, New York (1988)Google Scholar
  17. 17.
    Fillieger, R., Hongler, M.-O., Streit, L.: Connection between an exactly solvable stochastic optimal control problem and a nonlinear reaction-diffusion equation. J. Optimiz. Theory Appl. 137, 497–505 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.L. (ed.) Ècole d’Ètè de Probabilitès de Saint-Flour XV-XVII. Lecture Notes in Mathematics, vol. 1362, pp. 102–203. Springer, New York (1988)Google Scholar
  19. 19.
    Fortet, R.: Résolution d’un système d’equations de M. Schrödinger. J. Math. Pure Appl. IX, 83–105 (1940)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 323–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Léonard, C.: From the Schrödinger problem to the Monge-Kantorovich problem. J. Funct. Anal. 262, 1879–1920 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Léonard, C.: A survey of the Schroedinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4), 1533–1574 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Liang, S., Medich, D., Czajkowsky, D.M., Sheng, S., Yuan, J., Shao, Z.: Ultramicroscopy 84, 119 (2000)CrossRefGoogle Scholar
  24. 24.
    Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes. Probab. Theory Relat. Fields 129, 245–260 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116, 1815–1835 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Opt. 47(3), 1127–1139 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pavon, M., Wakolbinger, A.: On free energy, stochastic control, and Schroedinger processes. In: Di Masi, G.B., Gombani, A., Kurzhanski, A. (eds.) Modeling, Estimation and Control of Systems with Uncertainty, pp. 334–348. Birkauser, Boston (1991)Google Scholar
  28. 28.
    Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Villani, C.: Topics in Optimal Transportation, vol. 58. AMS, Providence (2003)zbMATHGoogle Scholar
  30. 30.
    Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)zbMATHGoogle Scholar
  31. 31.
    Vinante, A., Bignotto, M., Bonaldi, M., et al.: Feedback cooling of the normal modes of a massive electromechanical system to submillikelvin temperature. Phys. Rev. Lett. 101, 033601 (2008)CrossRefGoogle Scholar
  32. 32.
    Wakolbinger, A.: Schroedinger bridges from 1931 to 1991. In: Proceedings of the 4th Latin American Congress in Probability and Mathematical Statistics, Mexico City 1990, Contribuciones en probabilidad y estadistica matematica, 3, 61–79 (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadovaItaly

Personalised recommendations