Kantorovich Mass Transfer Research Articles

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.

The two reflector design problem for forming a flat wavefront from a point source as an optimal mass transfer problem

The article deals with a problem of calculating two reflecting surfaces that form a given irradiance distribution with a flat wavefront, provided that a point source of light is used. A notion of a weak solution for the said problem is formulated and the equivalence of this problem and the Monge–Kantorovich mass transfer is proven.

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.

General preferences and utility functions. An approach based on the dual Kantorovich problem

In this paper, with help of a special dual Kantorov� ich mass transfer problem new existence conditions are obtained for utility functions of general (nontotal and nontransitive) preference relations. Further, by a preference on a set X of alternatives we mean an arbitrary binary relation � (totality or transi� tivity in general is not required). If xy (in such a case, we will write also yx), then the alternative x is more preferable than the alternative y. We connect withtwo binary relations: the strict preference � (xy means that xy holds but yx fails) and the equal worth relation ~ (x ~ y means that both xy and yx hold true). Alternatives x and y are incomparable if none of them is more preferable than other, i.e. none of relations xy and yx is satisfied. A preference is called total if any two alternatives are comparable. A preference with properties of reflexivity (xx ∀x ∈ X) and transitivity (∀x, y, z ∈ X: xy, yz ⇒ xz) is called a preorder (or quasiordering in other termi� nology).

Wasserstein metric convergence method for Fokker–Planck equations with point controls

Monge–Kantorovich mass transfer theory is employed to obtain an existence and uniqueness result for solutions to Fokker–Planck Equations with time dependent point control. Existence for an approximate problem is established together with a convergence analysis in the Wasserstein distance through equivalence with weak- ⋆ convergence.

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 Solutions to the Mass Transfer Problem

This paper studies the Monge--Kantorovich mass transfer (MT) problem on metric spaces, with possibly unbounded cost function. Conditions are given under which the MT problem is solvable and, furthermore, an optimal solution can be obtained as the weak limit of a sequence of optimal solutions to suitably approximating MT problems.

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.

Variational Principle for General Diffusion Problems

We employ the Monge–Kantorovich mass transfer theory to study the existence of solutions for a large class of parabolic partial differential equations. We deal with nonhomogeneous nonlinear diffusion problems (of Fokker–Planck type) with time-dependent coefficients. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results of the authors, where proofs of convergence in the case of a right-hand side in the equation is given by these methods. To prove the existence of weak solutions we establish an interesting maximum principle for such equations. This involves comparison with the solution for the corresponding homogeneous, time-independent equation.

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.

Nonlocal Geometric Expansion of Convex Planar Curves

We consider a class of nonlocal geometric equations for expanding curves in the plane, arising in the study of evolutions governed by Monge–Kantorovich mass transfer. We construct convex solutions, given convex initial data. In order to obtain such solutions, we develop a new version of Perron's method. We give applications to the problem of characterizing fast/slow diffusion limits.

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.