Convergence framework for the second boundary value problem for the Monge-Ampère equation

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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-Ampère equation. Viscosity solutions are a powerful tool for analyzing 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-Ampère 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.

Original languageEnglish (US)
Pages (from-to)945-971
Number of pages27
JournalSIAM Journal on Numerical Analysis
Volume57
Issue number2
DOIs
StatePublished - 2019

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Convergence
  • Finite difference methods
  • Monge-Ampère equation
  • Second boundary value problem

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