Boundary value problems posed over thin solids are often amenable to a dimensional reduction in that one or more spatial dimensions may be eliminated from the governing equation. One of the popular methods of achieving dimensional reduction is the Kantorovich method, where based on certain a priori assumptions, a lower-dimensional problem over a ‘mid-element’ is obtained. Unfortunately, the mid...

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting som...

We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on IR and a reference distribution on the d-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth in the case of elliptical distributions, but, for more general distributions, diffe...

In this paper we extend the contraction mapping method to prove various existence and uniqueness properties of (self-similar) random fractal measures, and establish exponential convergence results for approximating sequences defined by means of the scaling operator. For this purpose we introduce a version of the Monge Kantorovich metric on the class of probability distributions of random measur...

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

The present paper introduces new classes of Stancu–Kantorovich operators constructed in the King sense. For these operators, we establish some convergence results, error estimations theorems and graphical properties approximation for considered, namely, that preserve test functions e0(x)=1 e1(x)=x, e2(x)=x2, as well e1(x)=x e2(x)=x2. class e2(x)=x2 is a genuine generalization introduced by Indr...

In this paper, we present some applications of the multivariate sampling Kantorovich operators Sw to seismic engineering. The mathematical theory of these operators, both in the space of continuous functions and in Orlicz spaces, show how it is possible to approximate/reconstruct multivariate signals, such as images. In particular, to obtain applications for thermographic images a mathematical ...

In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of ...

In his book on convex polytopes [2] A.D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in Rn+1, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature [18]...

A family of neural network operators of the Kantorovich type is introduced and their convergence studied. Such operators are multivariate, and based on certain special density functions, constructed through sigmoidal functions. Pointwise as well as uniform approximation theorems are established when such operators are applied to continuous functions. Moreover, also L p approximations are consid...