## 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 language | English (US) |
---|---|

Pages (from-to) | 945-971 |

Number of pages | 27 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - 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