Finite Difference Schemes for the Monge-Ampère equation
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Related Papers
- Wide stencil finite difference schemes for the elliptic Monge-Ampère 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)
- (with Jean-David Benamou and Brittany Froese) Finite Difference Schemes for the Elliptic Monge-Ampère Equation ESAIM: M2AN, 23 February 2010
- (with Brittany Froese) Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation
- (with Brittany Froese) Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
Short description of the work
The first numerical method is a wide stencil convergent difference scheme
The second numerical method is not provably convergent, but in practice performs as well, or better. 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.
The next paper builds a three dimensional convergent scheme, and computes the solution using Newton's method
The monotone scheme is used to build a more accurate hybrid scheme, which is also solved using Newton's method.
Abstract for Paper 3
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.
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