A rate of convergence of numerical optimal transport problem with quadratic cost

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PDE Seminar
April 15, 2021
1:50PM - 2:50PM
Location
Online: Zoom info below

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Add to Calendar 2021-04-15 13:50:00 2021-04-15 14:50:00 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 Speaker's URL:  https://sites.math.rutgers.edu/~wz222/ Abstract:  In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. 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 $c(x,y)$. 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 by introducing tools in computational geometry, such as Brun-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. URL associated with Seminar:  https://research.math.osu.edu/pde/ Additional Dates &/or Times Zoom Link: https://osu.zoom.us/j/6146883919?pwd=M2hHVnFaMjVMb1pYT2FFZGdEVDIwdz09 Meeting ID: 6146883919 Password: 314159 Location Zoom ID 6146883919 / Password 314159 Online: Zoom info below Department of Mathematics math@osu.edu America/New_York public
Description

Speaker:  Wujun Zhang (Rutgers University)

Title:  A rate of convergence of numerical optimal transport problem with quadratic cost

Speaker's URL:  https://sites.math.rutgers.edu/~wz222/

Abstract:  In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. 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 $c(x,y)$. 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 by introducing tools in computational geometry, such as Brun-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.

URL associated with Seminar:  https://research.math.osu.edu/pde/


Additional Dates &/or Times
Zoom Link: https://osu.zoom.us/j/6146883919?pwd=M2hHVnFaMjVMb1pYT2FFZGdEVDIwdz09
Meeting ID: 6146883919
Password: 314159

Location
Zoom ID 6146883919 / Password 314159

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