  We introduce a new overlapping Domain Decomposition Method (DDM) to solve the
fully nonlinear Monge-Amp\`ere equation. While DDMs have been extensively
studied for linear problems, their application to fully nonlinear partial
differential equations (PDE) remains limited in the literature. To address this
gap, we establish a proof of global convergence of these new iterative
algorithms using a discrete comparison principle argument. Several numerical
tests are performed to validate the convergence theorem. These numerical
experiments involve examples of varying regularity. Computational experiments
show that method is efficient, robust, and requires relatively few iterations
to converge. The results reveal great potential for DDM methods to lead to
highly efficient and parallelizable solvers for large-scale problems that are
computationally intractable using existing solution methods.
