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arXiv · 1807.04216

Convergence framework for the second boundary value problem for the Monge-Amp\`ere equation

Brittany Froese Hamfeldt
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Abstract

It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the second boundary value problem for the Monge-Amp\`ere equation. Viscosity solutions are a powerful tool for analysing and approximating fully nonlinear elliptic equations. However, we demonstrate that this nonlinear elliptic equation does not satisfy a comparison principle and thus existing convergence frameworks for viscosity solutions are not valid. We introduce an alternative PDE that couples the usual Monge-Amp\`ere equation to a Hamilton-Jacobi equation that restricts the transportation of mass. We propose a new interpretation of the optimal transport problem in terms of viscosity subsolutions of this PDE. Using this reformulation, we develop a framework for proving convergence of a large class of approximation schemes for the optimal transport problem. Examples of existing schemes that fit within this framework are discussed.

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Brittany Froese Hamfeldt. 2018-07-11. Convergence framework for the second boundary value problem for the Monge-Amp\`ere equation. https://arxiv.org/abs/1807.04216

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