Monge-Kantorovich Mass Transfer Problem Research Articles

A multiscale approach to a linear programming problem for calculating a refractive optical element forming a given far-field illumination distribution

We consider an approach to the calculation of a refractive optical element generating a prescribed irradiance distribution in the far field for a plane incident beam based on a certain variational problem. We consider an explicit formulation of this problem in the form of the Monge-Kantorovich mass transfer problem. We also demonstrate a connection between the mass transfer problem and the dual linear variational problem. A numerical solution of the linear variational problem is also considered. The direct solution of this type of problem presents a huge computational complexity. To overcome this difficulty we use the so-called multiscale approach based on constructing a chain of approximations that are solutions on refining grids.

Efficient Method to Solve the Monge–Kantarovich Problem Using Wavelet Analysis

In this paper, we present and justify a methodology to solve the Monge–Kantorovich mass transfer problem through Haar multiresolution analysis and wavelet transform with the advantage of requiring a reduced number of operations to carry out. The methodology has the following steps. We apply wavelet analysis on a discretization of the cost function level j and obtain four components comprising one corresponding to a low-pass filter plus three from a high-pass filter. We obtain the solution corresponding to the low-pass component in level j−1 denoted by μj−1*, and using the information of the high-pass filter components, we get a solution in level j denoted by μ^j. Finally, we make a local refinement of μ^j and obtain the final solution μjσ.

Support quadric method in non-imaging optics problems that can be reformulated as a mass transfer problem

The article deals with problems of generating desired illumination patterns, formulated in a special way. More precisely, we consider problems that can be reformulated as a Monge–Kantorovich mass transfer problem with some cost function. For all problems of this type, we uniformly formulate the support quadric method and show that it coincides with the gradient method for finding the maximum of a certain concave function.

Supporting quadric method for collimated beams

We consider the problem of calculating a refractive element with two surfaces, forming a flat front and a given distribution of illumination. The supporting quadrics method is formulated for calculating a given optical element and it is shown that this method coincides with the gradient method for some functional related to the problem of the Monge-Kantorovich mass transfer problem. This enables adaptive selection of the step in the supporting quadric method. At the end of the article a design example is given.

Unnormalized optimal transport

We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier in [4]. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampére type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm [6]. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [25] (which is a precursor of our work here) to have desirable properties.

An optimization problem with volume constraint with applications to optimal mass transport

In this manuscript we study the following optimization problem with volume constraint:min{1p∫Ω|∇v|pdx−∫∂ΩgvdHN−1:v∈W1,p(Ω), and LN({v>0})≤α}. Here Ω⊂RN is a bounded and smooth domain, g is a continuous function and α is a fixed constant such that 0<α<LN(Ω). Under the assumption that ∫∂Ωg(x)dHN−1>0 we prove that a minimizer exists and satisfies{−Δpup=0in {up>0}∪{up<0},|∇up|p−2∂up∂η=gon ∂Ω∩∂({up>0}∪{up<0}),LN({up>0})=α.Next, we analyze the limit as p→∞. We obtain that any sequence of weak solutions converges, up to a subsequence, limpj→∞⁡upj(x)=u∞(x), uniformly in Ω‾, and uniform limits, u∞, are solutions to the maximization problem with volume constraintmax⁡{∫∂ΩgvdHN−1:v∈W1,∞(Ω),‖∇v‖L∞(Ω)≤1 and LN({v>0})≤α}. Furthermore, we obtain the limit equation that is verified by u∞ in the viscosity sense. Finally, it turns out that such a limit variational problem is connected to the Monge-Kantorovich mass transfer problem with the involved measures are supported on ∂Ω and along the limiting free boundary, ∂{u∞≠0}. Furthermore, we show some explicit examples of solutions for certain configurations of the domain and data.

Parameter Estimation for Dielectric Media Variations Based on the FDTD Method and the Monge–Kantorovich Mass Transfer Problem

An efficient and consistent methodology is developed in this paper for the precise evaluation of unknown dielectric permittivity media parameters involved in the design of 3-D waveguide structures. In essence, the selected configuration is numerically solved for a set of predetermined permittivity values by means of a time-domain (herein, the finite-difference time-domain) technique at least twice, depending on the desired level of accuracy. Finally, implementing the Monge-Kantorovich mass transfer problem formulation, the required fields are rigorously evaluated for a broader range of values, as substantiated by the results obtained from various realistic arrangements.

Analytic solutions for the approximated 1-D Monge–Kantorovich mass transfer problems

This paper mainly investigates the approximation of a global maximizer of the 1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge-Kantorovich problem will be demonstrated.

Designing double freeform optical surfaces for controlling both irradiance and wavefront

We propose an improved double freeform-optical-surface design method for shaping a prescribed irradiance distribution whilst forming a desired wavefront from a given incident beam. This method generalizes our previous work [Opt. Exp. 21, 14728-14735 (2013)] to tackle non-separable beam irradiances. We firstly compute a proper ray mapping using an adaptive mesh method in the framework of the L2 Monge-Kantorovich mass transfer problem. Then, we construct the two freeform optical surfaces according to this mapping using a modified simultaneous point-by-point procedure which is aimed to minimize the surface errors. For the first surface, the modified procedure works by firstly approximating a value to the next point by only using the slope of the current point and then improving it by utilizing both slopes of the two points based on Snell's law. Its corresponding point on the second surface can be computed using the constant optical path length condition. A design example of producing a challenging irradiance distribution and a non-ideal wavefront demonstrates the effectiveness of the method.

An Optimization Problem for Mass Transportation with Congested Dynamics

Starting from the work by Brenier [Extended Monge–Kantorovich theory, in Optimal Transportation and Applications (Martina Franca 2001), Lecture Notes in Math. 1813, Springer-Verlag, Berlin (2003), pp. 91–121], where a dynamic formulation of mass transportation problems was given, we consider a more general framework, where different kinds of cost functions are allowed. This seems relevant in some problems presenting congestion effects as, for instance, traffic on a highway, crowds moving in domains with obstacles, and, in general, in all cases where the transportation does not behave as in the classical Monge setting. We show some numerical computations obtained by generalizing to our framework the approximation scheme introduced in Benamou and Brenier [A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375–393].

The Neumann problem for the [formula omitted]-Laplacian and the Monge–Kantorovich mass transfer problem

We consider the natural Neumann boundary condition for the ∞ -Laplacian. We study the limit as p → ∞ of solutions of − Δ p u p = 0 in a domain Ω with | D u p | p − 2 ∂ u p / ∂ ν = g on ∂ Ω . We obtain a natural minimization problem that is verified by a limit point of { u p } and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the Monge–Kantorovich mass transfer problems when the measures are supported on ∂ Ω .

On the consistency of the mass transfer problem

Conditions are given under which the Monge–Kantorovich mass transfer problem on general metric spaces and with unbounded cost function has a feasible solution.

Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

It is shown that the problem of designing a two- reflector system transforming a plane wave front with given in- tensity into an output plane front with prescribed output in- tensity can be formulated and solved as the Monge-Kantorovich mass transfer problem.

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

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 prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work “Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle” [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.

Strong duality of the Monge-Kantorovich mass transfer problem in metric spaces

This paper studies the Monge–Kantorovich mass transfer (MT) problem on metric spaces and with an unbounded cost function. Conditions are given under which the strong duality condition holds; that is, MT and its dual MT\(^*\) are both solvable and their optimal values coincide.

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Summary. The $L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

Differential equations methods for the Monge-Kantorovich mass transfer problem

Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.