Fast Bellman algorithm for real Monge-Ampere equation

TL;DR

This paper presents a new, faster numerical algorithm for solving the real Monge-Ampère equation by representing it as an infimum of linear operators and applying Bellman's principle, with proven convergence and superior performance.

Contribution

The paper introduces a novel Bellman-based numerical scheme for the Monge-Ampère equation, demonstrating improved speed and convergence over existing methods.

Findings

Algorithm is 3-10 times faster for smooth, convex cases.

Algorithm is 20-100 times faster for degenerate cases.

Proven convergence under certain conditions.

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.