A multigrid scheme for 3D Monge-Amp\`ere equations

TL;DR

This paper develops a fast multigrid solver for the 3D Monge-Ampère equation, enabling efficient solutions for applications like medical imaging, by combining a nonlinear Gauss-Seidel smoother with a multigrid approach.

Contribution

It introduces a nonlinear multigrid method with a convexity mechanism for solving 3D Monge-Ampère equations efficiently.

Findings

The finite difference scheme achieves second-order accuracy.

The multigrid solver significantly reduces computational time.

Numerical experiments confirm the method's effectiveness.

Abstract

The elliptic Monge-Amp\`ere equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation 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 proposed multigrid solver.