30 research outputs found
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
Monotone and Consistent discretization of the Monge-Ampere operator
We introduce a novel discretization of the Monge-Ampere operator,
simultaneously consistent and degenerate elliptic, hence accurate and robust in
applications. These properties are achieved by exploiting the arithmetic
structure of the discrete domain, assumed to be a two dimensional cartesian
grid. The construction of our scheme is simple, but its analysis relies on
original tools seldom encountered in numerical analysis, such as the geometry
of two dimensional lattices, and an arithmetic structure called the
Stern-Brocot tree. Numerical experiments illustrate the method's efficiency
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re
equation with second boundary value condition, provided the target is a convex
set. This yields a fast adaptive method to numerically solve the Optimal
Transport problem between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
A Monge-Ampère-solver for free-form reflector design
In this article we present a method for the design of fully free-form reflectors for illumination systems. We derive an elliptic partial differential equation of the Monge-Ampère type for the surface of a reflector that converts an arbitrary parallel beam of light into a desired intensity output pattern. The differential equation has an unusual boundary condition known as the transport boundary condition. We find a convex or concave solution to the equation using a state of the art numerical method. The method uses a nonstandard discretization based on the diagonalization of the Hessian. The discretized system is solved using standard Newton iteration. The method was tested for a circular beam with uniform intensity, a street light, and a uniform beam that is transformed into a famous Dutch painting. The reflectors were verified using commercial ray tracing software
A Monge-Ampère-solver for free-form reflector design
In this article we present a method for the design of fully free-form reflectors for illumination systems. We derive an elliptic partial differential equation of the Monge-Ampère type for the surface of a reflector that converts an arbitrary parallel beam of light into a desired intensity output pattern. The differential equation has an unusual boundary condition known as the transport boundary condition. We find a convex or concave solution to the equation using a state of the art numerical method. The method uses a nonstandard discretization based on the diagonalization of the Hessian. The discretized system is solved using standard Newton iteration. The method was tested for a circular beam with uniform intensity, a street light, and a uniform beam that is transformed into a famous Dutch painting. The reflectors were verified using commercial ray tracing software
Monotone and Consistent discretization of the Monge-Ampere operator
International audienceWe introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency