A multigrid scheme for 3D Monge-Ampère equations

The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community [Tad12, FGN13]. Fast three dimensional solvers are needed, for example in medical image registration, [HZTA04, HRT10, HPM+09, HM07], but are not yet available. We build fast solvers for smooth solutions in three dimensions, using a nonlinear fullapproximation storage multigrid method. Starting from a second-order accurate centered finite difference approximation, we present a nonlinear Gauss-Seidel iterative method which has a mechanism for selecting the convex solution of the equation. The iterative method is used as an effective smoother, combined with the full-approximation storage multigrid method. Numerical experiments are provided to validate the accuracy of the finite difference scheme and illustrate the computational efficiency of the multigrid algorithm. The solution time is almost linear in the number of variables. Problems of size 643 are solved in seconds and of size 1283 are solved in a couple of minutes on a recent model laptop.