A multigrid scheme for 3D Monge–Ampère equations^*

* The work described in this article is a result of a collaboration made possible by the IMA's New Directions Short Course on ‘Topics on Control Theory: Optimal Mass Transportation’ in June, 2014.

ABSTRACT

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. 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 centred 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.

KEYWORDS:

• Monge–Ampère equation
• nonlinear partial differential equations
• finite difference method
• Gauss–Seidel iteration
• FAS multigrid method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N06
• 65N22
• 65N55
• 65N12 (MSC2010)

Acknowledgments

The authors would like to thank the two anonymous referees for their valuable comments and suggestions that have greatly contributed to improving the original version of this manuscript. The first author gratefully acknowledges the support and hospitality provided by the IMA during his participation in the IMA's New Directions Short Course on ‘Topics on Control Theory’, which took place from May 27 to June 13, 2014.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

* The work described in this article is a result of a collaboration made possible by the IMA's New Directions Short Course on ‘Topics on Control Theory: Optimal Mass Transportation’ in June, 2014.

Additional information

Funding

This project was supported in part by Guangdong Province Science and Technology Development Grant (foreign cooperation project: 2013B051000075) of China, and in part by NSF 1021203, 1419028 of the United States.