TY - JOUR
T1 - A multigrid scheme for 3D Monge–Ampère equations*
AU - Liu, Jun
AU - Froese, Brittany D.
AU - Oberman, Adam M.
AU - Xiao, Mingqing
N1 - Funding Information:
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.
Publisher Copyright:
© 2016 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2017/9/2
Y1 - 2017/9/2
N2 - 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.
AB - 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.
KW - FAS multigrid method
KW - Gauss–Seidel iteration
KW - Monge–Ampère equation
KW - finite difference method
KW - nonlinear partial differential equations
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U2 - 10.1080/00207160.2016.1247443
DO - 10.1080/00207160.2016.1247443
M3 - Article
AN - SCOPUS:84996558197
SN - 0020-7160
VL - 94
SP - 1850
EP - 1866
JO - International Journal of Computer Mathematics
JF - International Journal of Computer Mathematics
IS - 9
ER -