Numerical Methods for Optimal Transport Problems
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Omarov, Daniyar
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Abstract
In my work, I present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I provide a solution for a discrete problem with an obstacle in the domain.
Next, I consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I introduce a new class of problems called separable, for which very accurate methods can be devised.
Lastly, I propose a novel implementation of Newton's method for solving semi-discrete OT problems for cost functions that are a positive combination of $p$-norms, $1<p<\infty$. By exploiting the geometry of the Laguerre cells, one can obtain an efficient and reliable implementation of Newton's method to find the sought network structure. Moreover, I discuss the generalization of the semi-discrete problem in higher dimensions and the application of the proposed algorithm in drawing electoral districts.
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Date
2024-04-17
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Dissertation