Year: 2026
Type: preprint
Abstract: In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Ampère Dirichlet problem on bounded convex planar domains. We approximate the prescribed probability measure by point clouds and construct the least concave tropical roof with zero boundary values and a corner at each point, using repeated updates of affine planes with integral gradients. For universally generic clouds of $N$ points in a fixed compact subset of the interior, convergence of the empirical measures implies uniform convergence of the roofs, divided by $\sqrt{N}$, to the Aleksandrov solution. The limiting measure may be singular. Euler's and Pick's formulas relate curvature to point counts, with an $O(N^{-1/2})$ discrepancy against compactly supported $C^1$ tests. A removal estimate gives almost-sure uniform convergence for independent samples from any absolutely continuous probability law on the whole domain. For generic clouds in a fixed interior compact set of a rational-slope polygon, suitably rescaled sandpile odometers converge to the same solution along meshes chosen sufficiently fine for each cloud. Their deficit measures converge to its negative Laplacian.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171659582