On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

Journal of Optimization Theory and Applications

Volumn 169, Issue 2, 2016, Pages 671-691

  Chen, Yongxin ^a   Georgiou, Tryphon T ^a   Pavon, Michele ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Optimal transport; Schr dinger bridge; Stochastic control

Indexed keywords

BROWNIAN MOVEMENT; COMPUTATION THEORY; FISHER INFORMATION MATRIX; FLUID DYNAMICS; STOCHASTIC CONTROL SYSTEMS; TRANSPORT PROPERTIES;

BRIDGE PROBLEMS; CONTROL PERSPECTIVE; FLUID-DYNAMICS; NOISE LIMIT; OPTIMAL TRANSPORT; PARALLEL A; SCHRÖDINGE BRIDGE; STOCHASTIC CONTROL; TRANSPORT PROBLEMS; VARIATIONAL PROBLEMS;

STOCHASTIC SYSTEMS;

EID: 84940494745     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-015-0803-z     Document Type: Article

References (59)

1. Monge, G.: Mémoire sur la théorie des déblais et des remblais, De l’Imprimerie Royale (1781)
2. Schrödinger, E.: Über die Umkehrung der Naturgesetze, Sitzungsberichte der Preuss. Phys. Math. Klasse. 10, 144–153 (1931)
3. Schrödinger, E.: Sur la théorie relativiste de l’electron et l’interpretation de la mécanique quantique. Ann. Inst. H. Poincaré. 2, 269 (1932)
4. Kantorovich, L.: On the transfer of masses. Dokl. Akad. Nauk. 37, 227–229 (1942)
5. Rachev, S., Rüschendorf, L.: Mass Transportation Problems: Theory, vol. 1. Springer, New York (1998)
6. Villani, C.: Topics in Optimal Transportation, vol. 58. AMS, Providence (2003)
7. Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, New York (2008)
8. Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996)
9. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)
10. Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numerische Mathematik 84(3), 375–393 (2000)
11. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Verlag, Basel (2008)
12. Ning, L., Georgiou, T.T., Tannenbaum, A.: Matrix-valued Monge–Kantorovich optimal mass transport. IEEE Trans. Autom. Control 60(2), 373–382 (2015)
13. Fortet, R.: Résolution d’un système d’equations de M. Schrödinger. J. Math. Pure Appl. 9, 83–105 (1940)
14. Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)
15. Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 323–331 (1975)
16. Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.L. (ed.) Ècole d’Ètè de Probabilitès de Saint-Flour XV–XVII, vol. 1362, pp. 102–203. Springer, New York (1988)
17. Wakolbinger, A.: Schrödinger bridges from 1931 to 1991. Contribuciones en probabilidad y estadistica matematica 3, 61–79 (1992)
18. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)
19. Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985)
20. Zambrini, J.C.: Stochastic mechanics according to E. Schrödinger. Phys. Rev. A 33(3), 1532–1548 (1986)
21. Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. Math. Phys. 27(9), 2307–2330 (1986)
22. Fenyes, I.: Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik. Z. Physik 132, 81–106 (1952)
23. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. 85, 166–179 (1952)
24. Bohm, D., Vigier, J.P.: Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys. Rev. 96, 208–216 (1954)
25. Levy, B.C., Krener, A.J.: Stochastic mechanics of reciprocal diffusions. J. Math. Phys. 37, 769 (1996)
26. Rosenbrock, H.H.: Doing quantum mechanics with control theory. IEEE Trans. Autom. Control 54, 73–77 (2000)
27. Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)
28. Dai Pra, P., Pavon, M.: On the Markov processes of Schrödinger, 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, pp. 497–504. Birkhäuser, Boston (1990)
29. Pavon, M., Wakolbinger, A.: On free energy, stochastic control, and Schrödinger processes. In: Di Masi, G.B., Gombani, A., Kurzhanski, A.A. (eds.) Modeling, Estimation and Control of Systems with Uncertainty, pp. 334–348. Birkhäuser, Boston (1991)
30. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4), 1533–1574 (2014)
31. Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Optim. 47(3), 1127–1139 (2008)
32. Pavon, M., Ticozzi, F.: Discrete-time classical and quantum Markovian evolutions: maximum entropy problems on path space. J. Math. Phys. 51, 042104–042125 (2010)
33. Léonard, C.: From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. 262, 1879–1920 (2012)
34. Georgiou, T.T., Pavon, M.: Positive contraction mappings for classical and quantum Schrödinger systems. J. Math. Phys. 56(033301), 1–24 (2015)
35. Angenent, S., Haker, S., Tannenbaum, A.: Minimizing flows for the Monge–Kantorovich problem. SIAM J. Math. Anal. 35(1), 61–97 (2003)
36. Filliger, R., Hongler, M.O.: Relative entropy and efficiency measure for diffusion-mediated transport processes. J. Phys. A: Math. Gen. 38, 1247–1250 (2005)
37. Filliger, R., Hongler, M.O., Streit, L.: Connection between an exactly solvable stochastic optimal control problem and a nonlinear reaction-diffusion equation. J. Optim. Theory Appl. 137, 497–505 (2008)
38. Chen, Y., Georgiou, T.T.: Stochastic bridges of linear systems. Preprint, http://arxiv.org/abs/1407.3421. IEEE Trans. Autom. Control (to appear)
39. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution, Part I. preprint, http://arxiv.org/abs/1408.2222. IEEE Trans. Autom. Control (to appear)
40. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution, Part II. Preprint, http://arxiv.org/abs/1410.3447. IEEE Trans. Autom. Control (to appear)
41. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system. Preprint, http://arxiv.org/abs/1502.01265 (submitted for publication)
42. Chen, Y., Georgiou, T.T., Pavon, M.: Entropic and displacement interpolation: a computational approach using the Hilbert metric. Preprint, http://arxiv.org/abs/1506.04255 (submitted for publication)
43. Chen, Y., Georgiou, T.T., Pavon, M.: Fast cooling for a system of stochastic oscillators. Preprint, http://arxiv.org/abs/1411.1323v2 (submitted for publication)
44. 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)
45. Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116, 1815–1835 (2006)
46. Carlen, E.: Stochastic mechanics: a look back and a look ahead. In: Faris, W.G. (ed.) Diffusion, Quantum Theory and Radically Elementary Mathematics, vol. 47, pp. 117–139. Princeton University Press, Princeton (2006)
47. Kosmol, P., Pavon, M.: Lagrange Lemma and the optimal control of diffusions II: nonlinear Lagrange functionals. Syst. Control Lett. 24(3), 215–221 (1995)
48. Fleming, W.H., Soner, M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006)
49. Tabak, E.G., Trigila, G.: Data-driven optimal transport. Commun. Pure. Appl. Math. doi:10.1002/cpa.21588 (2014)
50. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)
51. Föllmer, H.: Time Reversal on Wiener Space. Springer, New York (1986)
52. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)
53. Wakolbinger, A.: A simplified variational characterization of Schrödinger processes. J. Math. Phys. 30(12), 2943–2946 (1989)
54. Pavon, M.: Stochastic control and nonequilibrium thermodynamical systems. Appl. Math. Optim. 19, 187–202 (1989)
55. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)
56. Nagasawa, M.: Stochastic variational principle of Schrödinger Processes. In: Cinlar, E., Chung, K.L., Getoo, R.K., Fitzsimmons, P.J., Williams, R.J. (eds.) Seminar on Stochastic Processes, pp. 165–175. Birkhäuser, Boston (1989)
57. Pavon, M.: Hamilton’s principle in stochastic mechanics. J. Math. Phys. 36, 6774–6800 (1995)
58. Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Physik 40, 322–326 (1926)
59. Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of inertial particles diffusing anisotropically with losses. In: Proceedings of the American Control Conference (2015)