The Monge–Ampère equation

Optimal transport: discretization and algorithms

“…Dirichlet boundary conditions). We refer the reader to the pioneering work of Oliker-Prussner [79] and to the survey by Neilan, Salgado and Zhang [77].…”

“…These methods are able to solve optimal transport problems provided that the maximizer of (1.10) is a viscosity solution to the Monge-Ampère equation (1.11), imposing restrictions on its regularity. For the Monge-Ampère equation with Dirichlet conditions, we refer to the recent survey by Neilan, Salgado and Zhang [77].…”

Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces

“…Numerical resolution has been addressed quite recently by research. A review of is given in (Neilan, Salgado, and Zhang 2020).…”

Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation

“…The extension to strictly convex domains is however possible; either with the techniques from [20] or with other finite element approaches for curved domains, cf. [32] and the references therein.…”

“…Other existing finite element methods involve consistent linearization of (1.1), cf., e.g., [7,31,3], or the addition of small perturbations via the bi-Laplacian [13,15]. The survey article [32] provides a thorough overview on FDMs and FEMs for the numerical approximation of solutions to (1.1). We furthermore refer to the review articles [12,30] on more general fully nonlinear PDEs.…”