Monge-Kantorovich Problem Research Articles

Unbalanced and Partial $$L_1$$ L 1 Monge–Kantorovich Problem: A Scalable Parallel First-Order Method

We propose a new algorithm to solve the unbalanced and partial $$L_1$$L1-Monge---Kantorovich problems. The proposed method is a first-order primal-dual method that is scalable and parallel. The method's iterations are conceptually simple, computationally cheap, and easy to parallelize. We provide several numerical examples solved on a CUDA GPU, which demonstrate the method's practical effectiveness.

Adaptive approximation of the Monge–Kantorovich problemviaprimal-dual gap estimates

The Monge–Kantorovich problem arises as a special case for linear cost functionals in optimal transportation problems. It leads to a convex minimization problem with limited regularity properties. The convergent finite element discretization and iterative solution of the problem and its dual are addressed. Based on these approximations a computable upper bound for the primal-dual gap is derived which is suitable for efficient local mesh refinement. Numerical experiments reveal a significant improvement of related adaptive methods.

Variational approach to calculation of light field eikonal function for illuminating a prescribed region.

The problem of calculation of the light field eikonal function providing focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is obtained that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the source region (in which the eikonal function is defined) and a point of the target region. This result demonstrates that the sought-for eikonal function corresponds to a mapping, for which the total distance between the points of the original plane and the target region is minimized. The formalism proposed in the present work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. Besides, the calculation of the "ray mapping" corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed approach is illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangle and a beam of square section into a ring.

Lack of regularity of the transport density in the Monge problem

In this paper, we provide a family of counter-examples to the regularity of the transport density in the classical Monge–Kantorovich problem. We prove that the W1,p regularity of the source and target measures f± does not imply that the transport density σ is W1,p, that the BV regularity of f± does not imply that σ is BV and that f±∈C∞ does not imply that σ is W1,p, for large p.

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.

Solutions to multi-marginal optimal transport problems concentrated on several graphs

We study solutions to the multi-marginal Monge–Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution must concentrate on either finitely many or countably many graphs. We show that local differential conditions on the cost, known to imply local d -rectifiability of the solution, are sufficient to imply a local version of the first of our conditions. We exhibit two examples of cost functions satisfying our conditions, including the Coulomb cost from density functional theory in one dimension. We also prove a number of results relating to the uniqueness and extremality of optimal measures. These include a sufficient condition on a collection of graphs for any competitor in the Monge–Kantorovich problem concentrated on them to be extremal, and a general negative result, which shows that when the problem is symmetric with respect to permutations of the variables, uniqueness cannot occur except under very special circumstances.

A metaheuristic for a numerical approximation to the mass transfer problem

Abstract This work presents an improvement of the approximation scheme for the Monge-Kantorovich (MK) mass transfer problem on compact spaces, which is studied by Gabriel et al. (2010), whose scheme discretizes the MK problem, reduced to solve a sequence of finite transport problems. The improvement presented in this work uses a metaheuristic algorithm inspired by scatter search in order to reduce the dimensionality of each transport problem. The new scheme solves a sequence of linear programming problems similar to the transport ones but with a lower dimension. The proposed metaheuristic is supported by a convergence theorem. Finally, examples with an exact solution are used to illustrate the performance of our proposal.

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.

Optimal mass transport-based adaptive mesh method for phase-field models of two-phase fluid flows

In this paper, we present an efficient adaptive mesh method for solving phase-field model of a mixture of two-phase incompressible fluid flows. The adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation, also known as Monge–Kantorovich problem (MKP). The numerical solution of the MKP is computed by taking the gradient of the steady state solution of a parabolic Monge–Ampère equation (PMAE). Several numerical experiments are conducted to demonstrate the performance of the PMAE method for solving phase-field models. The numerical results illustrate that the adaptive mesh computed with PMAE method captures well the moving interface of the phase-field function.

On Properties of the Generalized Wasserstein Distance

The Wasserstein distances Wp (p \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W2 as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance W1(μ, ν) of two probability measures μ, ν and the supremum of the integrals in d(μ − ν) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances \({W_p^{a,b}}\), defined in terms of both the classical Wasserstein distance Wp and the total variation (or L1) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here p plays the same role as for the classic Wasserstein distance, while a and b are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between \({W_2^{a,b}}\) and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality a la Kantorovich–Rubinstein establishing the equality between \({W_1^{1,1}}\) and the flat metric.

The intrinsic dynamics of optimal transport

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a suciently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probility densities is unique.

On the Monge–Kantorovich problem with additional linear constraints

The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for c-monotonicity.

Numerical solution of the Monge–Kantorovich problem by density lift-up continuation

We present an iterative method to numerically solve the L² Monge-Kantorovich problem. The method is based on a Picard fixed point iteration of the linearized problem. Examples relative to the transport of two-dimensional densities show that the present method can significantly reduce the computational time over existing methods, especially when the Wasserstein distance between the densities is small.

A note on the Monge-Kantorovich problem in the plane

The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng propose a probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane into a Dirichlet boundary problem associated to a nonlinear elliptic equation. Their results are original and sound, however, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results.

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem.

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of N player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article, we emphasize the role of optimal transport theory in (i) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity and (ii) the analysis of Cournot-Nash equilibria.

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\). Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\)-convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\), we show that for every probability measure \(\mu \) on \(X\), there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \), and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\)-almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\), then \(S\) is necessarily a \(\mu \)-measure preserving involution (i.e., \(S^2=I\)) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \)-almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.

OPTIMIZATION OF THE COST FUNCTION IN THE MONGE-KANTOROVICH PROBLEM (MKP) UNDER THE MONGE CONDITION

This paper presents a method for adapting the cost function in the Monge–Kantorovich Problem (MKP) to a classification task. More specifically, we introduce a criterion that allows to learn a cost function which tends to produce large distance values for elements belonging to different classes and small distance values for elements belonging to the same class. Under some additional constraints (one of them being the well-known Monge condition), we show that the optimization of this criterion writes as a linear programming problem. Experimental results on synthetic data show that the output optimal cost function provides good retrieval performances in the presence of two types of perturbations commonly found in histograms. When compared to a set of various commonly used cost functions, our optimal cost function performs as good as the best cost function of the set, which shows that it can adapt well to the task. Promising results are also obtained on real data for two-class image retrieval based on grayscale intensity histograms.

Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

We address the problem of whether a bounded measurable vector field from a bounded domain Ω into \({\mathbb{R}^d}\) is N-cyclically monotone up to a measure preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain ΩN. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N − 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability measures on ΩN which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on a graph of the form x → (x, Sx, S 2 x,..., S N-1 x), where S is a μ-measure preserving transformation on Ω such that S N = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.

Decoupling of DeGiorgi-Type Systems via Multi-Marginal Optimal Transport

We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [7] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [13]. Orientable nonlinearities and compatible cost functions turn out to be also related to submodular functions, which appear in rearrangement inequalities of Hardy-Littlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.

Monge problem in metric measure spaces with Riemannian curvature-dimension condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N<∞. For the first marginal measure, we assume that μ0≪m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value.Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.