Many-point tropical relaxation and the Monge--Ampère equation
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 √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 C1 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.
Related
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- Continuous solutions to the quaternionic Monge-Ampère equation on locally flat HKT manifolds
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BibTeX
@misc{kalinin2026manypoint,
title = {{Many-point tropical relaxation and the Monge--Ampère equation}},
author = {Nikita Kalinin and Ernesto Lupercio and Higinio Serrano and Mikhail Shkolnikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.25878},
eprint = {2607.25878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25878},
}
RIS
TY - UNPB
TI - Many-point tropical relaxation and the Monge--Ampère equation
AU - Nikita Kalinin
AU - Ernesto Lupercio
AU - Higinio Serrano
AU - Mikhail Shkolnikov
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.25878
UR - https://arxiv.org/abs/2607.25878
AB - 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 √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.
ER -