# A Numerical Method for the Elliptic Monge-Ampère Equation with Transport Boundary Conditions

@article{Froese2012ANM, title={A Numerical Method for the Elliptic Monge-Amp{\`e}re Equation with Transport Boundary Conditions}, author={Brittany D. Froese}, journal={SIAM J. Sci. Comput.}, year={2012}, volume={34} }

The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\`ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary condition. In this paper, we propose a method for solving the transport problem by…

## Figures and Tables from this paper

## 87 Citations

Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher

- MathematicsSIAM J. Numer. Anal.
- 2011

This article builds a wide stencil finite difference discretization for the Monge-Ampere equation and proves convergence of Newton's method and provides a systematic method to determine a starting point for the Newton iteration.

Numerical Solution of the Optimal Transportation Problem via viscosity solutions for the Monge-Ampere equation

- Mathematics
- 2012

A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local…

Numerical solution of the second boundary value problem for the Elliptic Monge-Amp ere equation

- Mathematics
- 2012

This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampere equation. The boundary conditions correspond to the optimal transportation of measures supported on…

A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2019

The methodology discussed in this article is able to handle domains with curved boundaries and unstructured meshes, using piecewise affine continuous approximations, while preserving optimal, or nearly optimal, convergence orders for the approximation error.

Solving the Monge–Ampère equations for the inverse reflector problem

- Mathematics
- 2015

The inverse reflector problem arises in geometrical nonimaging optics: given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light…

Numerical Methods for Hamilton-Jacobi-Bellman Equations with Applications

- Computer Science
- 2019

This thesis proposes a deep neural network framework for the HJB equations emerging from the study of American options in high dimensions, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer.

Convergent Filtered Schemes for the Monge-Ampère Partial Differential Equation

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2013

This article establishes a convergence result for filtered schemes, which are nearly monotone, and employs this framework to construct a formally second-order scheme for the Monge--Ampere equation and presents computational results on smooth and singular solutions.

A Least-Squares Method for Optimal Transport Using the Monge-Ampère Equation

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2015

A novel numerical method to solve the problem of optimal transport and the related elliptic Monge--Ampere equation is introduced, one of the few numerical algorithms capable of solving this problem efficiently with the proper transport boundary condition.

A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation

- Mathematics, Computer Science
- 2014

This work introduces a new formulation that couples the viscosity and Aleksandrov solution definitions and shows that it is equivalent to the original problem, and describes a local reformulation of the subgradient measure at the Diracs that leads to a consistent, monotone discretisation of the equation.

Solving the Monge-Ampère equation on triangle-meshes for use in optical freeform design

- Computer ScienceOptical Systems Design
- 2018

This work proposes a method to solve the Monge-Ampère equation on convex bounded domains by using triangle meshes and by minimizing the difference between prescribed and actual target light distribution which is computed by tracing rays through the optical surface.

## References

SHOWING 1-10 OF 68 REFERENCES

Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher

- MathematicsSIAM J. Numer. Anal.
- 2011

This article builds a wide stencil finite difference discretization for the Monge-Ampere equation and proves convergence of Newton's method and provides a systematic method to determine a starting point for the Newton iteration.

Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation

- MathematicsJ. Comput. Phys.
- 2011

Moving Mesh Generation Using the Parabolic Monge--Amp[e-grave]re Equation

- Computer ScienceSIAM J. Sci. Comput.
- 2009

This method gives a new technique for performing $r-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension).

An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions.

- Mathematics
- 2006

In this article, we discuss the numerical solution of the Dirichlet problem for the real elliptic MongeAmpère equation, in two dimensions, by an augmented Lagrangian based iterative method. To derive…

An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization

- Computer ScienceJ. Comput. Phys.
- 2008

Mixed Finite Element Methods for the Fully Nonlinear Monge-Ampère Equation Based on the Vanishing Moment Method

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2009

A family of Hermann-Miyoshi-type mixed finite element methods for approximating the solution of the regularized fourth-order problem, which computes simultaneously $u^\varepsilon$ and the moment tensor $\sigma^ \varePSilon:=D^2u^⩽𝕂𝕽$, is developed.

An Efficient Numerical Method for the Solution of the L2 Optimal Mass Transfer Problem

- Computer ScienceSIAM J. Sci. Comput.
- 2010

A new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L(2) mass transport mapping is presented, employing a direct variational method.

Two Numerical Methods for the elliptic Monge-Ampère equation

- Mathematics
- 2010

The numerical solution of the elliptic Monge-Ampere Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and…

On the second boundary value problem for Monge-Ampère type equations and optimal transportation

- Mathematics
- 2006

This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of…

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

- Mathematics
- 2003

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with…