Robust and fast iterative method for the elliptic Monge-Amp\`ere equation

Abstract

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal transport, geometric optics, and differential geometry. The proposed method linearises the equation and uses a fixed-point iteration (L-scheme), solving a Poisson problem in each step with a weighted residual as the right-hand side. This algorithm is robust against discretisation, nonlinearities, and degeneracies. For a weight greater than the largest eigenvalue of the Hessian, contraction in $H^2$ and $L^\infty$ is proven for both classical and generalised solutions, respectively. The method's performance can be enhanced by using preconditioners or Green's functions. Test cases demonstrate that the scheme outperforms Newton's method in speed and stability.