A numerical method for the elliptic monge-amperè equation with transport boundary conditions

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Abstract

The problem of optimal mass transport arises in numerous applications, including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary condition. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Amp ere equation with Neumann boundary conditions. To enable mappings between variable densities, we extend an earlier discretization of the equation to allow for right-hand sides that depend on gradients of the solution [B. D. Froese and A. M. Oberman, SIAM J. Numer. Anal., 49 (2011), pp. 1692-1714]. This discretization provably converges to the viscosity solution. The resulting system is solved efficiently with Newton's method. We provide several challenging computational examples that demonstrate the effectiveness and efficiency (O(M)-O(M 1.3) time) of the proposed method.

Original languageEnglish (US)
Pages (from-to)A1432-A1459
JournalSIAM Journal on Scientific Computing
Volume34
Issue number3
DOIs
StatePublished - 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Elliptic partial differential equations
  • Finite difference methods
  • Monge-Amp ere equation
  • Optimal transport
  • Viscosity solutions

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