Optimal Mass Transport for Registration and Warping
International Journal of Computer Vision
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Scientific Computing
Space deformations, surface deformations and the opportunities in-between
Journal of Computer Science and Technology
SIAM Journal on Numerical Analysis
Moving Mesh Generation Using the Parabolic Monge-Ampère Equation
SIAM Journal on Scientific Computing
The Monge-Ampère equation: Various forms and numerical solution
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Quadratic Finite Element Approximations of the Monge-Ampère Equation
Journal of Scientific Computing
Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation
Journal of Computational Physics
Hi-index | 31.45 |
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Ampere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three-dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.