TY - JOUR

T1 - Convergence framework for the second boundary value problem for the Monge-Ampère equation

AU - Hamfeldt, Brittany Froese

N1 - Funding Information: ∗Received by the editors July 23, 2018; accepted for publication (in revised form) April 3, 2019; published electronically April 30, 2019. http://www.siam.org/journals/sinum/57-2/M120191.html Funding: The work of the author was supported by National Science Foundation grants DMS-1619807 and DMS-1751996. †Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 (bdfroese@njit.edu). Funding Information: The work of the author was supported by National Science Foundation grants DMS-1619807 and DMS-1751996. Publisher Copyright: © 2019 Society for Industrial and Applied Mathematics

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Convergence

KW - Finite difference methods

KW - Monge-Ampère equation

KW - Second boundary value problem

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U2 - https://doi.org/10.1137/18M1201913

DO - https://doi.org/10.1137/18M1201913

M3 - Article

VL - 57

SP - 945

EP - 971

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 2

ER -