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Optimal transport and Schrödinger bridges from a control and computational perspective

The dynamic version of the Monge-Kantorovich optimal mass transport (OMT) inf π∈Π(μ,ν) ∫ R×R c(x, y)dπ(x, y), which is available whenever the cost functionn c(x, y) derives from a Lagrangian action, has, in a suitable sense the Schrödinger bridge problem (SBP) as a regular approximation (Mikami 2004, Mikami-Thieullen 2006,2008, Léonard 2012, 2014). The latter, originating with Schrödinger in 1931 and with important contributions due to Fortet, Beurlin, Jamison and Föllmer, is a maximum entropy problem connected to large deviations of the empirical distribution on path-space. It has important applications, for instance, to cooling of micro and macro mechanical systems [5]. Recently, we have derived implementable forms of the solution for general linear stochastic systems [2], [3], [4]. For Markov chains and Kraus maps of statistical quantum mechanics implementable solutions have been presented in [1]. In this talk, based on joint work with Yongxin Chen and Tryphon Georgiou, University of Minnesota, I wish to unveil some profound connections between the two theories (and their generalizations) which allow, in particular, to exploit the implementable solutions of SBP to get approximate solutions of a general OMT problem.