Convergence of a least-squares splitting method for the Monge-Amp\`ere equation

TL;DR

This paper proves the linear convergence of a nonlinear least-squares splitting method for solving the Monge-Ampère equation on the 2D torus, providing the first rigorous convergence analysis for this approach.

Contribution

It offers the first rigorous convergence proof for the splitting method applied to the Monge-Ampère equation in the periodic setting, using a geometric reformulation as an alternating-projection scheme.

Findings

Proves linear convergence of the method in H^2 norm for initial data close to a solution.

Establishes the transversality of tangent spaces at the solution, ensuring contraction.

Provides a functional-analytic explanation for the numerical robustness of the method.

Abstract

We study the theoretical convergence of the nonlinear least-squares splitting method for the Monge-Amp\`ere equation in which each iteration decouples the pointwise nonlinearity from the differential operator and consists of a local nonlinear update followed by the solution of two sequential Poisson-type elliptic problems. While the method performs well in computations, a rigorous convergence theory has remained unavailable. We observe that the iteration admits a reformulation as an alternating-projection scheme on Sobolev spaces $H^m$, $m\ge 0$. At a solution, the G\^ateaux differentials of the projection maps are the linear projections onto the corresponding tangent spaces. We prove that these tangent spaces are transverse, and hence the linearization of the alternating-projection map is a contraction by classical Hilbert-space theory for alternating projections. Building on this geometric characterization, we prove linear convergence in $H^2$ of the splitting method on the two-dimensional torus $\mathbb{T}^2$ for initial data sufficiently close to a solution $u\in H^4$. To the best of our knowledge, this yields the first rigorous convergence result for this splitting method in the periodic setting and provides a functional-analytic explanation for its observed numerical robustness.