Related papers: The Monge-Ampere equation: various forms and numer…
A method for the reconstruction of the primordial density fluctuation field is presented. Various previous approaches to this problem rendered {\it non-unique} solutions. Here, it is demonstrated that the initial positions of dark matter…
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating…
In many scientific fields imaging is used to relate a certain physical quantity to other dependent variables. Therefore, images can be considered as a map from a real-world coordinate system to the non-negative measurements being acquired.…
We present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear…
We introduce an integral representation of the Monge-Amp\`ere equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs…
We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the…
A Monge-Amp\`ere (MA) equation arises when seeking an optimally transported mesh that equidistributes a given monitor function in Cartesian space. This MA equation is a fully nonlinear PDE, with a source term that is a function of the…
This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal…
We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an…
We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation. Error estimates are given for Lagrange…
We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.
The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and…
We study Monge-Amp\`ere gravity (MAG) as an effective theory of cosmological structure formation through optimal transport theory. MAG is based on the Monge-Amp\`ere equation, a nonlinear version of the Poisson equation, that relates the…
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an…
In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…
We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space…
This is an introduction to a particular class of auxiliary complex Monge-Amp\`ere equations which had been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential…
Monge-Ampere gravitation is a nonlinear modification of classical Newtonian gravitation, when the Monge-Ampere equation substitutes for the Poisson equation. We establish, through two applications of the large deviation principle, that the…
We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove…