Convergence of a least-squares splitting method for the Monge-Ampère equation

We study the theoretical convergence of the nonlinear least-squares splitting method for the Monge-Ampère 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 Gâteaux 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.

💡 Research Summary

The paper addresses a long‑standing open problem: providing a rigorous convergence analysis for the nonlinear least‑squares splitting method introduced for the Monge–Ampère equation. The Monge–Ampère equation, det D²u = f, is a fully nonlinear second‑order elliptic PDE whose solutions are required to be convex. Classical Galerkin methods are not directly applicable because the equation lacks a weak formulation in standard Sobolev spaces. The splitting method, originally proposed in