## Abstract

A numerical method for the solution of the elliptic Monge-Ampère Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem, is presented. A local representation of the OT boundary conditions is combined with a finite difference scheme for the Monge-Ampère equation. Newton's method is implemented, leading to a fast solver, comparable to solving the Laplace equation on the same grid several times. Theoretical justification for the method is given by a convergence proof in the companion paper [4]. Solutions are computed with densities supported on non-convex and disconnected domains. Computational examples demonstrate robust performance on singular solutions and fast computational times.

Original language | English (US) |
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Pages (from-to) | 107-126 |

Number of pages | 20 |

Journal | Journal of Computational Physics |

Volume | 260 |

DOIs | |

State | Published - Mar 1 2014 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## Keywords

- Convexity
- Finite difference methods
- Fully nonlinear elliptic partial differential equations
- Monge Ampère equation
- Monotone schemes
- Numerical methods
- Optimal Transportation
- Viscosity solutions