2012/08/23 by Jean‐David Benamou, Benamou, Jean-David, Brittany D. Froese +3 · 1 citation
Engineering · Mathematics · Social Sciences · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Optimization and Mathematical Programming #Transportation Planning and Optimization
paper · pdf · doi:10.48550/arxiv.1208.4870
openalex publication_date 2012/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A numerical method for the solution of the elliptic Monge-Ampere Partial\nDifferential Equation, with boundary conditions corresponding to the Optimal\nTransportation (OT) problem is presented. A local representation of the OT\nboundary conditions is combined with a finite difference scheme for the\nMonge-Ampere equation. Newton's method is implemented leading to a fast solver,\ncomparable to solving the Laplace equation on the same grid several times.\nTheoretical justification for the method is given by a convergence proof in the\ncompanion paper (Benamou et al., 2012). In this paper, the algorithm is\nmodified to a simpler compact stencil implementation and details of the\nimplementation are given. Solutions are computed with densities supported on\nnon-convex and disconnected domains. Computational examples demonstrate robust\nperformance on singular solutions and fast computational times.\n