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arXiv · 2505.04370

Fast Bellman algorithm for real Monge-Ampere equation

Aleksandra Le·Frank Wikström
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Abstract

In this paper, we introduce a new numerical algorithm for solving the Dirichlet problem for the real Monge--Ampere equation. The idea is to represent the non-linear Monge--Ampere operator as an infimum of a class of linear elliptic operators and use Bellman's principle to construct a numeric scheme for approximating the operator attaining this infimum. Moreover, we prove convergence of the proposed algorithm (under suitable technical assumptions) and discuss its strengths and weaknesses. We also demonstrate the performance of the method on several examples with various degrees of regularity and degeneracy and compare the results to two existing methods. Our method runs considerably faster than the ones used for comparison, improving the running time by a factor of 3--10 for smooth, strictly convex examples, and by a factor of 20--100 or more for mildly degenerate examples.

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Aleksandra Le, Frank Wikström. 2026-03-12. Fast Bellman algorithm for real Monge-Ampere equation. https://arxiv.org/abs/2505.04370

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