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

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.