The Monge-Ampere equation: various forms and numerical methods

Abstract

We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampere problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampere equation is a sum of such distributions.

24 pages, 2 tables, 5 figures, 32 references. Submitted to J. Computational Physics. Times of runs added, multiple improvements of the manuscript implemented.

References in corpus (3)

• The Monge-Ampère-Kantorovich approach to reconstruction in cosmology
• Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation
• Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method

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• Spline element method for the Monge-Ampere equation
• Uniform limit of discrete convex functions
• Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
• Iterative scheme for solving optimal transportation problems arising in reflector design