Global Approximation Properties Research Articles

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

The objective of this paper is to construct univariate and bivariate blending type α-Schurer–Kantorovich operators depending on two parameters α∈0,1 and ρ>0 to approximate a class of measurable functions on 0,1+q,q>0. We present some auxiliary results and obtain the rate of convergence of these operators. Next, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s K-functional in different function spaces. Moreover, we present some study on numerical and graphical analysis for our operators.

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

In this article, we introduce generalized beta extension of Szacute{a}sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.

α-Schurer Durrmeyer operators and their approximation properties

The key goal of the present research article is to introduce a new sequence of linear positive operator i.e., α-Schurer Durrmeyer operator and their approximation behaviour on the basis of function η (z), where η infinitely differentiable on [0,1], η(z)=0, η(1)=1 and η'(z)>0, for all z∈[0,1]. Further, we calculate central moments and basic estimates for the sequence of the operators. Moreover, we discuss the rate of convergence and order of approximation in terms of modulus of continuity, smoothness, Korovkin theorem, and Peeter's K-functional. Lastly, local and global approximation properties are studied in the subsequent section.

Beltrami fields and knotted vortex structures in incompressible fluid flows

AbstractThis paper gives a survey on recent results about the existence of knotted vortex structures in incompressible fluids. This includes the proof of Lord Kelvin's conjecture on the existence of knotted vortex tubes in steady Euler flows and a new probabilistic approach to address Arnold's speculation that typical Beltrami fields should exhibit vortex lines of arbitrary topological complexity. We review the key tools to establish these results: the global approximation properties of Beltrami fields, the inverse localization principle in spectral theory, and the theory of Gaussian random Beltrami fields, which permits the introduction of probabilistic considerations in the picture. Finally, we include some applications of these techniques to other problems, such as the existence of vortex reconnections for smooth solutions of the 3D Navier–Stokes equation and the evolution of local hot spots under the heat equation.

Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions.

Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

<p style='text-indent:20px;'>The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a} $\end{document}</tex-math></inline-formula>sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.</p>

Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

In this paper, we introduce a novel extension of the Bernstein-Kantorovich-Stancu type operator of degree n with the help of multiple shape parameters. Voronovskaja and Grüss-Voronovskaja type approximation theorems are examined via Ditzian-Totik moduli of smoothness. We investigate basic statistical convergence properties with respect to a non-negative regular summability matrix. Moreover, using Ditzian-Totik moduli, local and global approximation properties associated to the proposed operator have been established. Finally, several illustrative examples are presented to demonstrate the efficiency, applicability and validity of the operator. The graphical and numerical results verify that the proposed operator gives better approximation as well as expand the previous Bernstein-Kantorovich type modifications including single parameter.

Parallel two-phase methods for global optimization on GPU

Developing general global optimization algorithms is a difficult task, specially for functions with a huge number of local minima in high dimensions. Stochastic metaheuristic algorithms can provide the only alternative for the solution of such problems since they are aimed at guaranteeing global optimality. However, the main drawback of these algorithms is that they require a large number of function evaluations in order to skip/discard local optima, thus exhibiting a low convergence order and, as a result, a high computational cost. Furthermore, the situation can become even worse with the increase of dimension. Usually the number of local minima highly increases, as well as the computational cost of the function evaluation, thus increasing the difficulty for covering the whole search space. On the other hand, deterministic local optimization methods exhibit faster convergence rates, requiring a lower number of functions evaluations and therefore involving a lower computational cost, although they can get stuck into local minima. A way to obtain faster global optimization algorithms is to mix local and global methods in order to benefit from higher convergence rates of local ones, while retaining the global approximation properties. Another way to speedup global optimization algorithms comes from the use of efficient parallel hardware architectures. Nowadays, a good alternative is to take advantage of graphics processing units (GPUs), which are massively parallel processors and have become quite accessible cheap alternative for high performance computing. In this work a parallel implementation on GPUs of some hybrid two-phase optimization methods, that combine the metaheuristic Simulated Annealing algorithm for finding a global minimum, with different local optimization methods, namely a conjugate gradient algorithm and a version of Nelder–Mead method, is presented. The performance of parallelized versions of the above hybrid methods are analyzed for a set of well known test problems. Results show that GPUs represent an efficient alternative for the parallel implementation of two-phase global optimization methods.

Adaptive fuzzy fault‐tolerant control for a class of unknown non‐linear dynamical systems

This study investigates the adaptive fuzzy fault-tolerant control (FTC) problem for a class of non-linear multi-input multi-output systems with interconnections among subsystems and external unknown disturbance. The introduced multiple coupling terms have unknown non-linear forms relying on the system outputs and the corresponding subsystems, which cause that the system is no longer pure-feedback. Moreover, the considered actuator failures have a lock-in-place type and a type of loss of effectiveness. To solve the problem of actuator faults, compensative adaptive laws are constructed by taking advantages of the global fuzzy approximation property and a novel backstepping design procedure. According to the Lyapunov stability theorem, it is shown that all the signals in the closed-loop system are bounded and the outputs can converge to a giving compact set. In addition, simulation examples are presented to validate the effectiveness of the proposed adaptive fuzzy FTC protocol. Compared with previous researches on non-linear systems with unknown coupling terms, the proposed adaptive fuzzy FTC approach is more general, which may provide promising prospects of implementing in practical industrial systems.

On $$({\varvec{p,q}})$$ ( p , q ) Baskakov–Durrmeyer–Stancu Operators

In the present paper, we introduce the generalized form of (p, q) Baskakov–Durrmeyer Operators with Stancu type parameters. We derived the local and global approximation properties of these operators and obtained the convergence rate and behaviour for the Lipschitz functions. Moreover, we give comparisons and some illustrative graphics for the convergence of operators to some function.

Approximation of Schurer type q-Bernstein-Kantorovich operators

In this paper, a kind of Schurer type q-Bernstein-Kantorovich operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are also studied by means of the modulus of continuity and the help of functions of the Lipschitz class. Then, the global approximation property is given for these operators.

Convergence properties of Ibragimov-Gadjiev-Durrmeyer operators

The purpose of the present paper is to study the local and global direct approximation properties of the Durrmeyer type generalization of Ibragimov Gadjiev operators defined in [Aral, A. and Acar, T., On Approximation Properties of Generalized Durrmeyer Operators, (submitted)]. The results obtained in this study consist of Korovkin type theorem which enables us to approximate a function uniformly by new Durrmeyer operators, and estimate for approximation error of the operators in terms of weighted modulus of continuity. These results are obtained for the functions which belong to weighted space with polynomial weighted norm by new operators which act on functions defined on the non compact interval [0.∞). We finally present a direct approximation result.

Enhancing Least-Squares Finite Element Methods Through a Quantity-of-Interest

In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-of-interest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest. We establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions. Central to the approach is that it offers an element-level estimate of the error in the quantity-of-interest. As a result, we introduce an adaptive approach that yields efficient, adaptively refined approximations. Several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology. Notably, the results show that the new approach is effective at improving the accuracy per total computational cost.

A Stable Nonlinear in Parameter Neural Network Controller for a Class of Saturated Nonlinear Systems

This paper deals with the tracking problem for a class of uncertain nonlinear systems subjected to actuator saturation constraint. Despite most of proposed control schemes for saturated systems which employed linear in parameter neural networks (LPNNs), in the present work nonlinear in parameter neural network (NLPNN) is introduced to support global approximation property. To compensate the effect of the input saturation constraint an auxiliary system is introduced and the error dynamics are modified based on the auxiliary states. Then, learning rules are achieved based on the back propagation (BP) algorithm and by adding two robustifying terms to the standard BP learning rules the stability of the overall system is ensured via Lyapunov direct method. Finally simulations performed on a “generalized pendulum” nonlinear system to illustrate the effectiveness of the proposed tracking control scheme.

Approximation theorems for q-Bernstein-Kantorovich operators

In the present paper we introduce a q-analogue of the Bernstein-Kantorovich operators and investigate their approximation properties. We study local and global approximation properties and Voronovskaja type theorem for the q-Bernstein-Kantorovich operators in case 0 < q < 1.

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h l?+?k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.

Global approximation properties of modified SMK operators

In this paper, introducing a general modification of the classical Sz?sz-Mirakjan-Kantorovich (SMK) operators, we study their global approximation behavior. Some special cases are also presented. 2010 Mathematics Subject Classifications. 41A25, 41A36. .

On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems

In this paper, we apply the coupling of local discontinuous Galerkin and boundary element methods to solve a class of non-linear exterior transmission problems in the plane. As a model, we consider a non-linear elliptic equation in an annular polygonal domain coupled with the Poisson equation in the surrounding unbounded exterior region. In addition, we assume discontinuous transmission conditions on the interface boundary. Our approach constitutes an extension to non-linear problems of the a priori error analysis developed recently for linear exterior transmission problems. Most of the techniques employed here are similar to the linear case but some differences appear. In particular, because of the non-homogeneous transmission conditions, the so-called numerical fluxes need to be suitably defined. We prove stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Then, we apply local and global approximation properties of the discrete spaces to obtain the a priori error estimates in the energy norm.

A circuit framework for robust manifold learning

Manifold learning and nonlinear dimensionality reduction addresses the problem of detecting possibly nonlinear structure in high-dimensional data and constructing lower-dimensional configurations representative of this structure. A popular example is the Isomap algorithm which uses local information to approximate geodesic distances and adopts multidimensional scaling to produce lower-dimensional representations. Isomap is accurate on a global scale in contrast to many competing methods which approximate locally. However, a drawback of the Isomap algorithm is that it is topologically instable, that is, incorrectly chosen algorithm parameters or perturbations of data may drastically change the resulting configurations. We propose new methods for more robust approximation of the geodesic distances using a viewpoint of electric circuits. In this way, we achieve both the stability of local methods and the global approximation property of global methods, while compromising with local accuracy. This is demonstrated by a study of the performance of the proposed and competing methods on different data sets.

An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods

In this paper we analyze the coupling of local discontinuous Galerkin (LDG) and boundary element methods as applied to linear exterior boundary value problems in the plane. As a model problem we consider a Poisson equation in an annular polygonal domain coupled with a Laplace equation in the surrounding unbounded exterior region. The technique resembles the usual coupling of finite elements and boundary elements, but the corresponding analysis becomes quite different. In particular, in order to deal with the weak continuity of the traces at the interface boundary, we need to define a mortar-type auxiliary unknown representing an interior approximation of the normal derivative. We prove the stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Finally, we apply local and global approximation properties of the subspaces involved to obtain the a priori error estimate in the energy norm.