We recall the history of the transportation (Kantorovich) metric and the Monge–Kantorovich problem. We also describe several little-known applications: the first one concerns the theory of decreasing sequences of partitions (tower of measures and iterated metric), the second one relates to Ornstein's theory of Bernoulli automorphisms (¯ d-metric), and the third one is the formulation of the str...

This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting which the velocity field is Sobolev regular spatial variable. estimate formulated with help of Kantorovich--Rubinstein distances logarithmic cost functions. Second, are extended to fields whose gradients singular integrals $L^1$ functions entailing new well-p...

In this study, the elastic bending of sector graphene sheet has been studied based on elasticity using Eringen Nonlocal Elasticity Theory. In order to do this, the balance equations governing the sector graphene sheet have been solved in terms of displacements with regard to nonlocal relationship of stress, shear theory of the first order, and obtained linear strains using developed Kantorovich...

We propose a constructive approach to solve the Monge-Kantorovich problem for chains of infinite order on a finite alphabet with an additive cost function. From this constructive description of the Kantorovich coupling we obtain, for any ǫ > 0, a perfect simulation algorithm for sampling from an ǫ-approximating coupling which assigns to the cost function an expectation which is ǫ-close to the m...

We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. In th...

Solutions to Monge-Kantorovich equations, expressing optimality condition in mass transportation problem with cost equal to distance, are stationary points of a critical-slope model for sand surface evolution. Using a dual variational formulation of sand model, we compute both the optimal transport density and Kantorovich potential as t → ∞ limit of evolving sand flux and sand surface, respecti...

The research is supported by Zhejiang Provincial Education Department Research Projects (Y201016421) Abstract We achieve that the optimal solutions according to Kantorovich-Rubinstein-Wasserstein Lp−distance (p > 2) (abbreviation: KRW Lp−distance) in a bounded region of Euclidean plane satisfy a partial differential equation. We can also obtain the similar results about Monge-Kantorovich proble...

In this paper, the well known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numeric...