The Monge-Ampere equation: various forms and numerical methods

TL;DR

This paper introduces three new forms of the Monge-Ampere equation, emphasizing a Fourier integral form with proven positivity and bounds, leading to a novel numerical method for solving space-periodic problems with applications in cosmology.

Contribution

The paper develops a new Fourier integral form of the Monge-Ampere equation and a corresponding numerical method with proven properties, advancing computational approaches in this area.

Findings

Established positivity and bounds for the Fourier integral kernels.

Developed a convergent numerical method for space-periodic Monge-Ampere problems.

Demonstrated method effectiveness on a cosmological test problem.

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.