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 introduced in [1] 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 [2,3], a rigorous convergence theory has remained unavailable. Our main observation is 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 develop two distinct strategies to establish local convergence of the original splitting method. The first applies on the two-dimensional torus $\mathbb{T}^2$ for initial data sufficiently close to a solution $u\in H^4(\mathbb{T}^2)$. The second applies to smooth bounded and convex domains for solutions $u\in C^4(\overlineΩ)$. To the best of our knowledge, this yields the first rigorous convergence result for the splitting method at the continuous level and provides a functional-analytic explanation for its observed numerical robustness.