Approximation Properties of the Blending-Type Bernstein–Durrmeyer Operators

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.

Stancu generalized Baskakov operators using inverse Polya–Eggenberger distribution for a real valued bounded function on $$[0,\infty )$$ and a non-negative real number $$\alpha $$ (which may depend on n). Dhamija and Deo (Appl Math Comput 286:15–22, 2016) introduced a Durrmeyer type modification of these generalized operators and studied the uniform convergence, the rate of convergence by means of the moduli of continuity and the Peetre’s K-functional. The purpose of the present paper is to continue to study the approximation properties of these Durrmeyer type operators. Local and global direct approximation theorems and a Voronovskaya type asymptotic theorem are established. The quantitative Voronovskaya and Gruss Voronovskaya type theorems are investigated by calculating the indispensible sixth order moment. The weighted approximation properties and the approximation of functions with derivatives of bounded variation are also studied.

In this note, we study approximation properties of a family of linear positive operators and establish asymptotic formula, rate of convergence, local approximation theorem, global approximation theorem, weighted approximation theorem, and better approximation for this family of linear positive operators.

In this paper, a generalized sequence of Baskakov operators connected to the Riemann–Liouville fractional integral is introduced. These sequences of operators deal with smooth approximation behavior in a wider class, i.e., a class of measurable functions in the Lebesgue sense. Further, estimates in terms of test functions and central moments are established. Using the weighted Korovkin-type theorem, uniform convergence in weighted spaces is studied. Additionally, uses of the classical modulus of continuity and Peetre’s K-functional to establish local and global direct approximation theorems are introduced. Lastly, the statistical convergence properties of the suggested operators are investigated. In the last section, theoretical results are supported by numerical examples with graphical and numerical illustrations.

Weighted isogeometric collocation based on Spline Projectors

Advances in computing technology to improve the automation of industrial production processes has grown the expectations for food products of high quality and safety standards. Computer vision and machine learning techniques provide an alternative for developing automated and cost-effective methods to accomplish the requirements in food industry. In this paper, we propose several computer vision algorithms to characterize the process of transformation of bakery goods during production. Features are extracted from images acquired from a camera mounted in the furnace of a manufacturing company that produces bakery foods. Radial Basis Function Neural Network (RBFNN) is used to model the transformation of appearance of a particular product of interest. The RBF Neural Network offers several advantages compared to other neural network architectures especially when a characterization of a transitional process is needed. RBFNN possesses the property of best approximation and the output of the network can be optimized by setting suitable values of the center and the spread of RBF. Experimental results indicate that gradient based features have higher recognition rate due to their ability to capture the texture.

Best approximation on a finite set by a non‐linear family of functions with respect to a general sum “norm”, which includes as a special case the Lp norms (1 < p < ∞), is considered. Properties of best approximations are given. It is shown that a local minimum of the error may not be a global minimum. Computation of best approximations is studied. Computational experience with approximation by a first degree polynomial plus an exponential with respect to the 3/2 power norm is discussed.

The choice of function space allows us to make conclusions in the multidimensional case that are analogous to results in the theory of spline functions of one variable. We establish the minimum norm property, the existence and uniqueness of a solution of the interpolation problem, the property of best approximation, and the convergence of interpolation processes.

In this paper, we introduce the Bezier variant of two new families of generalized Bernstein type operators. We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. By means of construction of suitable functions and the method of Bojanic and Cheng, we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

When solving differential equations by means of a Galerkin approach, the approximating spaces are not only supposed to have good approximation properties, but also they must allow easy and fast computations. In addition, if the goal is the development of a multilevel method to detect and follow local singularities, or a multigrid scheme to solve the resulting linear systems, then hierarchical bases are necessary. As an example, we mention the finite element bases which have been widely used over the last three decades.

Chapters 2 and 3 have shown that neural networks can be employed to identify dynamic systems. The main advantages of neural networks over conventional identification methods include simplicity of implementation and good approximation properties [Warwick et aI, 1992]. In feedforward network based identification schemes, neural networks are used to represent the implied static mapping between the available input and output data. The network structures (number of layers and number of units in each layer) are predefined and remain unchanged both during and after training. Successful identification is often dependent on proper pre-estimation of the network structure.

The character of convection-dominated, singularly perturbed boundary value problems requires their special numerical treatment in order to guarantee stability and resolve existing layers with acceptable accuracy. In addition to discretization methods particularly developed for this aim, recently more and more attention has been directed towards adapted triangulations of the computational domain. In this paper, an adaptive strategy based on an anisotropic refinement is developed for finite element methods. Starting from some a priori information about the location of layers, the so-called hybrid meshes are constructed. By these meshes, the flexibility of unstructured meshes, good approximation properties in layers, and relatively simple rules for a posteriori anisotropic refinement are combined with each other. The efficiency of this procedure is demonstrated by selected numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.

We introduce a new model of parametric contours defined in a continuous fashion. Our curve model relies on Hermite spline interpolation and can easily generate curves with sharp discontinuities; it also grants direct access to the tangent at each location. With these two features, the Hermite snake distinguishes itself from classical spline-snake models and allows one to address certain bioimaging problems in a more efficient way. More precisely, the Hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image. Using the formalism of spline theory, the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties. Finally, the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our Hermite snake model, and its usefulness is illustrated on real biological images acquired using brightfield, phase-contrast, differential-interference-contrast, and scanning-electron microscopy.

In this paper, we show that the integro quintic splines can locally be constructed without solving any systems of equations. The new construction does not require any additional end conditions. By virtue of these advantages the proposed algorithm is easy to implement and effective. At the same time, the local integro quintic splines possess as good approximation properties as the integro quintic splines. In this paper, we have proved that our local integro quintic spline has superconvergence properties at the knots for the first and third derivatives. The orders of convergence at the knots are six (not five) for the first derivative and four (not three) for the third derivative.

In this work we investigate the advantages of multiscale methods in Petrov–Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space, which only contains negligible fine scale information. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov–Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG continuous and a discontinuous Galerkin finite element multiscale method. Furthermore, we demonstrate that the Petrov–Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov–Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley–Leverett equation. To achieve this, we couple a PG discontinuous Galerkin finite element method with an upwind scheme for a hyperbolic conservation law.