Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams

We introduce a monotone (degenerate elliptic) discretization of the Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded. Our approach enjoys the simplicity of the Wide Stencil method, but significantly improves its accuracy using ideas from discretizations of optimal transport based on power diagrams. We establish the global convergence of a damped Newton solver for the discrete system of equations. Numerical experiments, in three dimensions, illustrate the scheme efficiency

Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions

We address the discretization of optimization problems posed on the cone of convex functions, motivated in particular by the principal agent problem in economics, which models the impact of monopoly on product quality. Consider a two dimensional domain, sampled on a grid of N points. We show that the cone of restrictions to the grid of convex functions is in general characterized by N^2 linear inequalities; a direct computational use of this description therefore has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of discrete convex functions, associated to stencils which can be adaptively, locally, and anisotropically refined. Numerical experiments optimize the accuracy/complexity tradeoff through the use of a-posteriori stencil refinement strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2. Modifications are anecdotic.

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure

Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance

We consider a two player game, where a first player has to install a surveillance system within an admissible region. The second player needs to enter the the monitored area, visit a target region, and then leave the area, while minimizing his overall probability of detection. Both players know the target region, and the second player knows the surveillance installation details.Optimal trajectories for the second player are computed using a recently developed variant of the fast marching algorithm, which takes into account curvature constraints modeling the second player vehicle maneuverability. The surveillance system optimization leverages a reverse-mode semi-automatic differentiation procedure, estimating the gradient of the value function related to the sensor location in time N log N

Minimal geodesics along volume preserving maps, through semi-discrete optimal transport

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the flow dimension is higher than the domain dimension, a striking and unavoidable consequence of this model. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the first time in two space dimensions.Comment: 21 pages, 9 figure

Monotone and Consistent discretization of the Monge-Ampere operator

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency

Greedy bisection generates optimally adapted triangulations

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.Comment: 24 page

Worst case and average case cardinality of strictly acute stencils for two dimensional anisotropic fast marching

We study a one dimensional approximation-like problem arising in the discretization of a class of Partial Differential Equations, providing worst case and average case complexity results. The analysis is based on the Stern-Brocot tree of rationals, and on a non-Euclidean notion of angles. The presented results generalize and improve on earlier work