Data driven optimal transportation and its application

The goal of this thesis is to propose a linear programming based numerical method to solve the data-driven optimal transport problem. Optimal transportation is widely applied in fields such as econometrics, image processing, shape optimization and many other areas. This thesis focuses on the numerical solution to this problem. Different methods for solving optimal transportation problem are briefly discussed and compared. A data driven formulation of the problem is presented and solved by an adaptive methodology, using refining meshes to decompose the problem into a sequence of finite linear programming problems. The formulation does not involve any density estimation; it imposes both local mass constraints and the local first moment constraints, which result in a better transport plan in terms of marginal density constraints than previous approaches that use the mass alone. The series of linear programming problems utilize at each level the solution from the previous coarser mesh to restrict the size of the function space where solutions are sought while guaranteeing the feasibility, accuracy and fast calculation of the new problem. The Wasserstein barycenter problem derives from optimal transportation; it is receiving growing attention due to its connection with multi marginal optimal transportation problems and the explanation of variability in data. The linear programming approach for pairwise optimal transportation is combined with an iterative scheme, to form a data driven algorithm for the Wasserstein barycenter problem. The algorithm keeps the benefit of our optimal transport solver and is very well suited to parallel computing. Numerical experiments and applications for both pairwise optimal transport problem and Wasserstein barycenter demonstrate the power and efficiency of this method