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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Num. Math. 84(3), 375–393 (2000)
Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)
Blaquière, A.: Controllability of a Fokker-Planck equation, the Schrödinger system, and a related stochastic optimal control. Dyn. Control 2(3), 235–253 (1992)
Georgiou, T.T., Pavon, M.: Positive contraction mappings for classical and quantum Schrödinger systems, May 2014, arXiv:1405.6650v2. J. Math. Phys. to appear
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
Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of inertial particles diffusing anisotropically with losses, arXiv:1410.1605v1, ACC Conf. (2015)
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
Chen, Y., Georgiou, T.T., Pavon, M.: Fast cooling for a system of stochastic oscillators, Nov 2014, arXiv:1411.1323v1
Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint, arXiv:1412.4430v1
Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system, arXiv:1502.01265v1
Chen, Y., Georgiou, T.T., Pavon, M.: A computational approach to optimal mass transport via the Schrödinger bridge problem (2015, in preparation)
Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)
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)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Applied Math., vol. 38, 2nd edn. Springer, New York (1998)
Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, New York (1988)
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)
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)
Fortet, R.: Résolution d’un système d’equations de M. Schrödinger. J. Math. Pure Appl. IX, 83–105 (1940)
Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 323–331 (1975)
Léonard, C.: From the Schrödinger problem to the Monge-Kantorovich problem. J. Funct. Anal. 262, 1879–1920 (2012)
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)
Liang, S., Medich, D., Czajkowsky, D.M., Sheng, S., Yuan, J., Shao, Z.: Ultramicroscopy 84, 119 (2000)
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)
Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116, 1815–1835 (2006)
Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Opt. 47(3), 1127–1139 (2008)
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)
Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57 (2002)
Villani, C.: Topics in Optimal Transportation, vol. 58. AMS, Providence (2003)
Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Chen, Y., Georgiou, T., Pavon, M. (2015). Optimal Mass Transport over Bridges. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-25040-3_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25039-7
Online ISBN: 978-3-319-25040-3
eBook Packages: Computer ScienceComputer Science (R0)