Applied and Computational Math Seminar: A rate of convergence of numerical optimal transport problem with quadratic cost

Speaker: Wujun Zhang, Rutgers University
Title: A rate of convergence of numerical optimal transport problem with quadratic cost

Abstract: The goal in optimal transportation is to transport a measure mu(x) into a measure nu(y) with minimal total effort with respect to a given cost function c(x,y). In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. On way to approximate the optimal transport solution is to approximate the measure mu by the convex combination of Dirac measure mu_h on equally spaced nodal set and solve the discrete optimal transport between mu_h and nu. If the cost function is quadratic, i.e. c(x,y) = |x-y|^2, the optimal transport mapping is related to an important concept from computational geometry, namely Laguerre cells. In this talk, we study the rate of convergence of the discrete optimal mapping using tools in computational geometry, such as Brunn-Minkowski inequality. We show that the rate of convergence of the discrete mapping measured in W^1_1 norm is of order O(h^2) under suitable assumptions on the regularity of the optimal mapping. We will also discuss the rate of convergence in the case that the optimal mapping is degenerate. 

Time: Friday, April 19, 2019, 1:30-2:30pm

Place: Exploratory Hall, Room 4106