Related papers: The second boundary value problem for a discrete M…
The Monge-Amp\`{e}re equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the…
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 consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed…
The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…
To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of…
In this paper we continue the analysis of the two-scale method for the Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete…
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…
These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Amp\`ere and linearized Monge-Amp\`ere equations. The…
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…
We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Amp\`ere equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function…
We formulate an Alternating Direction Method of Mul-tipliers (ADMM) that systematically distributes the computations of any technique for optimizing pairwise functions, including non-submodular potentials. Such discrete functions are very…
In this paper we study nonlinear second-order differential inclusions involving the ordinary vector $p$-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying…
In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both…
We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform…
We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed…
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their…
We address the numerical solution via Galerkin type methods of the Monge-Amp\`ere equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully…
This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is…
The aim of this paper is to obtain quantitative bounds for solutions to the optimal matching problem in dimension two. These bounds show that up to a logarithmically divergent shift, the optimal transport maps are close to be the identity…
Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of…