Original Optimal transport via L1 minimization

      In 1781, Monge proposed a distance function among histograms. We develop fast methods for it using techniques borrowed in L1 compressive sensing. Our method is very simple, easy to parallelize and can be combined with other regularizations. We use it in applications including partial optimal transport, image segmentation, image alignment and others. It is flexible enough to easily deal with histograms and other features of the data.
    • Optimal flux among two images I
    • Optimal flux among two images II
    • Partial optimal transport
    • Image segmentation I
    • Image segmentation II

    Optimal transport and Schrodinger type problems

      In 1966, Nelson derived Schrödinger equation by diffusion process. Nowadays this approach connects with the theory of optimal transport. We consider similar matters on finite graphs. We propose a discrete Schrödinger related equations from Nelson’s idea and optimal transport. The proposed equation enjoys several dynamical features.
    • Computed transportation from digits 4 to digits 1.

    Optimal Control and Robotics

      We design a new fast algorithm for a class of optimal control problems with constraints on both state and control variables. Instead of searching global minimizer(s) from all feasible paths, we consider the subset of paths with structure of optimal paths. By leveraging these paths, we transfer optimal control problems to a set of finite and different dimensional optimization problems with constraints. Moreover, for each of these finite dimensional subproblems, we apply methods from stochastic differential equations in order to find numerically all possible global minimizers of our original optimal control problem.
    • Froger game.
    • Finite drones' dynamical games.
    • Game theory via optimal transport

        We propose a new dynamic framework for finite or infinite player (population) discrete strategy games. By utilizing tools from optimal transportation theory, we derive Fokker-Planck equations of games. Furthermore, we introduce an associated Best-Reply Markov process that models players’ myopicity, greedy and uncertainty when making decisions. The model gives rise to a method to rank/select equilibria for both potential and non-potential games.
      • Constraint Rock-Paper-Scissors and its invariant distribution.