Monge-Ampere equation
Finite Difference Schemes for the Monge-Ampere equation
The first numerical method is a wide stencil convergent difference scheme
- Wide stencil finite difference schemes for the elliptic Monge-Ampere equation and functions of the eigenvalues of the Hessian, Discrete and Continuous Dynamical Systems series B (DCDS B) Volume 10, Number 1, July 2008 (221-238)
The second numerical method is not provably convergent, but in practice performs as well, or better,
- (with Jean-David Benamou and Brittany Froese) Finite Difference Schemes for the Elliptic Monge-Ampere Equation
This paper also has a discussion of non-convergent schemes, including an illustration of how many of these schemes perform better on smooth solutions, but are orders of magnitude slower when the solutions become more singular.
Numerical Methods for Optimal Transportation
- Have done the linear programming for general cost via approximation of measure by atoms.
- Planned: solving the quadratic cost problem/Monge-Ampere equation using the Legendre Transform.
Notes related to optimal transportation
See