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

  title={A Least-Squares Method for Optimal Transport Using the Monge-Amp{\`e}re Equation},
  author={C. R. Prins and Ren{\'e} Beltman and J. H. M. ten Thije Boonkkamp and Wilbert L. IJzerman and Teus W. Tukker},
  journal={SIAM J. Sci. Comput.},
In this article we introduce a novel numerical method to solve the problem of optimal transport and the related elliptic Monge--Ampere equation. It is one of the few numerical algorithms capable of solving this problem efficiently with the proper transport boundary condition. The computation time scales well with the grid size and has the additional advantage that the target domain may be nonconvex. We present the method and several numerical experiments. 
The second boundary value problem for a discrete Monge-Ampere equation with symmetrization
This work proposes a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport and uses a recently proposed scheme based on a partial discrete analogue of a symmetrization of the subdifferential.
A Newton Div-Curl Least-Squares Finite Element Method for the Elliptic Monge–Ampère Equation
A new finite element approach for the efficient approximation of classical solutions of the elliptic Monge–Ampère equation is developed, using an outer Newton-like linearization and a first-order system least-squares reformulation at the continuous level to define a sequence of first- order div-curl systems.
A continuation multiple shooting method for Wasserstein geodesic equation
This algorithm is based on use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem.
Viscosity subsolutions of the second boundary value problem for the Monge-Amp\`ere equation
It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the second boundary value problem for the Monge-Amp\`ere equation. Viscosity solutions are a powerful
Convergence Framework for the Second Boundary Value Problem for the Monge-Ampère Equation
It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the second boundary value problem for the Monge--Ampere equation. Viscosity solutions are a powerfu...
A Convergent Quadrature Based Method For The Monge-Ampère Equation
An integral representation of the Monge-Amp`ere equation is introduced, which leads to a new new method based upon numerical quadrature, which is monotone and immediately integrated into existing convergence proofs for themonotone equation with either Dirichlet or optimal transport boundary conditions.
A convergence framework for optimal transport on the sphere
We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the
A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design
Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampere type
A fast approach to optimal transport: the back-and-forth method
This work presents an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs, with an approximately exponential convergence rate.


An Efficient Numerical Method for the Solution of the L2 Optimal Mass Transfer Problem
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.
A Numerical Method for the Elliptic Monge-Ampère Equation with Transport Boundary Conditions
This paper proposes a method for solving the transport problem by iteratively solving a Monge-Amp\`ere equation with Neumann boundary conditions and extends an earlier discretization of the equation to allow for right-hand sides that depend on gradients of the solution.
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampere equation, Pucci’s Maximal and Minimal
On the second boundary value problem for Monge-Ampère type equations and optimal transportation
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
Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher
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.
Convergent Filtered Schemes for the Monge-Ampère Partial Differential Equation
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.
Minimizing Flows for the Monge-Kantorovich Problem
This work derives a novel gradient descent flow for the computation of the optimal transport map (when it exists) in the Monge-Kantorovich framework, and studies certain properties of the flow, including weak solutions as well as short- and long-term existence.
A Monge-Ampère-Solver for Free-Form Reflector Design
A method is presented for the design of fully free-form reflectors for illumination systems that converts an arbitrary parallel beam of light into a desired intensity output pattern using an elliptic partial differential equation of the Monge--Ampere type.