Kantorovich Type Research Articles

A Differential Model for Growing Sandpiles on Networks

We consider a system of differential equations of Monge--Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton--Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535--545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.

Some smoothness properties of the Lupaş-Kantorovich type operators based on Pólya distribution

In the present article, we study some smoothness properties of new Lupa?-Kantorovich type operators based on P?lya distribution, as uniform convergence and asymptotic behavior. In order to get the degree of approximation, some quantitative type theorems will be established. The bivariate extension of these operators, with some indispensable results will be also presented.

Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting

AbstractThe theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform andLpnorms, for continuous andLpfunctions, respectively. The above family of operators, are based upon kernels generated by sigmoidal functions. Multivariate approximation by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of sigmoidal functions for which the above theory holds have been presented.

Dunkl generalization of Kantorovich type Szász–Mirakjan operators via q-calculus

In this paper, we construct Kantorovich type Szász–Mirakjan operators generated by Dunkl generalization of the exponential function via [Formula: see text]-integers. We obtain some approximation results via well-known Korovkin’s type theorem for these operators and study convergence properties by using the modulus of continuity. Furthermore, we obtain the rate of convergence in terms of the classical, second-order, and weighted modulus of continuity.

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton–Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted ℓ1 Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values.

Approximation theorems for Kantorovich type Lupaș-Stancu operators based on \(q\)-integers

In this paper, we introduce a Kantorovich generalization of q-Stancu-Lupa¸s operators and investigate their approximation properties. The rate of convergence of these operators are obtained by means of modulus of continuity, functions of Lipschitz class and Peetre's K-functional. We also investigate the convergence of the operators in the statistical sense and give a numerical example in order to estimate the error in the approximation.

Approximation by Kantorovich form of modified Szász–Mirakyan operators

In the present article, we consider the Kantorovich type generalized Szász–Mirakyan operators based on Jain and Pethe operators [32]. We study local approximation results in terms of classical modulus of continuity as well as Ditzian–Totik moduli of smoothness. Further we establish the rate of convergence in class of absolutely continuous functions having a derivative coinciding a.e. with a function of bounded variation.

Kantorovich duality for general transport costs and applications

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. We also provide explicit examples of discrete measures satisfying the weak transport-entropy inequalities derived here.

Several Inequalities for Khatri-Rao Products of Hilbert Space Operators

We establish several inequalities for Khatri-Rao products of Hilbert space operators, involving ordinary products, ordinary powers, ordinary inverses, and Moore-Penrose inverses. Kantorovich type inequalities concerning Khatri-Rao products are also investigated. Our results generalize some matrix inequalities in the literature. In our case, we must impose some mild conditions on operators such as the closeness of their ranges. Furthermore, we develop new operator inequalities by using block partitioning technique and unital positive linear maps.

Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

The purpose of this paper is to introduce a family of q‐Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well‐known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and the second‐order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q‐Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.

$L^p$-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejér kernel

By use of the so-called max-product method, in this paper we associate to the truncated linear sampling operators based on the Fej\'er-type kernel, nonlinear sampling operators of Kantorovich type, for which we prove convergence results in the $L^{p}$-norm, $1\le p\le +\infty $, with quantitative estimates.

About the analogy between optimal transport and minimal entropy

We describe some analogy between optimal transport and the Schrodinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role.

Approximation properties of (p, q)-Bernstein type operators

Abstract We introduce a new generalization of the q-Bernstein operators involving (p, q)-integers, and we establish some direct approximation results. Further, we define the limit (p, q)-Bernstein operator, and we obtain its estimation for the rate of convergence. Finally, we introduce the (p, q)-Kantorovich type operators, and we give a quantitative estimation.

Approximation by k-th order modifications of Szász—Mirakyan operators

In this paper, we study the k-th order Kantorovich type modication of Szász—Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szász—Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.

Convergence for a family of neural network operators in Orlicz spaces

In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function f, are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.

Approximation by Kantorovich Type q-Bernstein-Stancu Operators

In this paper, we introduce a Kantorovich type generalization of q-Bernstein-Stancu operators. We study the convergence of the introduced operators and also obtain the rate of convergence by these operators in terms of the modulus of continuity. Further, we study local approximation property and Voronovskaja type theorem for the said operators. We show comparisons and some illustrative graphics for the convergence of operators to a certain function.

Approximation by Max-Product Neural Network Operators of Kantorovich Type

In the present paper, we develop the theory of max-product neural network operators in a Kantorovich-type version, which is suitable in order to study the case of L p -approximation for not necessarily continuous data. Moreover, also the case of the pointwise and uniform approximation of continuous functions is considered. Finally, several examples of sigmoidal functions for which the above theory can be applied are presented.

Some approximation properties of Kantorovich variant of Chlodowsky operators based on q-integer

In this paper, we introduce two digerent Kantorovich type generalization of the q Chlodowsky operators. For the first operators we give some weighted approximation theorems and a Voronovskaja type theorem. Also, we present the local approximation properties and the order of convergence forunbounded functions of these operators . For second operators, we obtain aweighted statistical approximation property

On (p, q)-generalization of Szász-Mirakyan Kantorovich operators

In this paper, we introduce Szasz-Mirakyan Kantorovich type of operators based on (p, q)-calculus. Using Krovokin’s type theorem, we show that operator converges uniformly. In second section, we study rate of convergence of operator using modulus of continuity and Peetre’s K-functional. In last section, we give Voronovskaya type results for operator.