Saturation of Cao–Gonska operators Λ* and Λ+

We give characterizations by K-functionals and moduli of smoothness for discretized versions of Cao–Gonska operators having O(n ^?2) as their saturation order.We also identify the corresponding saturation class.     

Maximal Abstract Monotonicity and Generalized Fenchel’s Conjugation Formulas

In this paper, we present a generalization of a theory of monotone operators in the framework of abstract convexity. We show that how generalized Fenchel’s conjugation formulas can be used to obtain some results on maximal abstract monotonicity. We give a necessary and sufficient condition for maximality of abstract monotone operators representable by abstract convex functions by using an additivity constraint qualification.     

Admissibility conditions for a class of quadrature formulas with random nodes

The present note is concerned with the theory of the Monte Carlo method. We consider the problem of the admissibility of quadrature formulas with random nodes. Based on the formulation of an admissibility condition we obtain a new class of such formulas.     

A new generalization of complex Stancu operators

In this paper, we investigate approximation properties of the complex form of an extension of the Bernstein polynomials, defined by Stancu by means of a probabilistic method. We obtain quantitative upper estimates for simultaneous approximation and the exact order of approximation by these operators attached to analytic functions in closed disks. Also, we prove that the new generalized complex Stancu operators preserve the univalence, starlikeness, convexity, and spirallikeness in the unit disk.     

Differentiated shift-invariant integral operators,univariate case

In a recent paper [1] the author, along with H. Gonska, introduced some wavelet type integral operators over the whole real line and studied their properties such as shift-invariance, global smoothness preservation, convergence to the unit, and preservation of probability distribution functions. These operators are very general and they are introduced through a convolution-like iteration of another general operator with a scaling type function. In this paper the author provides sufficient conditions, so that the derivatives of the above operators enjoy the same nice properties as their originals. A sufficient condition is also given so that the “global smoothness preservation” related inequality becomes sharp. At the end several applications are given, where the derivatives of the very general specialized operators are shown to fulfill all the above properties. In particular it is shown that they preserve continuous probability density functions.     

Further Results for kth order Kantorovich Modification of Linking Baskakov Type Operators

In this paper we consider a non-trivial link between Baskakov type operators and their genuine Durrmeyer type modification as well as the kth order Kantorovich variant. Recursion formulas for the moments and the images of monomials are proved in order to derive asymptotic expansions. Furthermore we investigate convexity properties of the linking operators and the limiting behavior for certain function spaces.     

Convex and coconvex-probabilistic wavelet approximation

Continuous functions are approximated by wavelet operators. These preserve convexity and r-convexity and transform continuous probability distribution functions into probability distribution functions at the same time preserving certain convexity conditions. The degree of this approximation is estimated by establishing some Jackson type inequalities     

On a Problem of Gonska

This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.     

Quadrature and orthogonal rational functions

Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence.     

ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION

Optimization problems with L¹-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l¹-regularization optimization,the resulting discretized L¹norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L¹-norm.In this paper,a new discretized scheme for the L¹-norm is presented.Compared to the new discretized scheme for L¹-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L¹-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.     

On the regularization of projection methods for solving III-posed problems

Summary In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.     

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori estimate of the norm for the infinite-dimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.     

On the convergence of the vortex loop method with regularization for the Neumann boundary value problem on a plane screen

We consider a linear integral equation with a supersingular integral treated in the sense of the Hadamard finite value, which arises in the solution of the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a doublelayer potential. We consider the case in which the exterior boundary value problem is solved outside a plane surface (a screen). For the integral operator in the above-mentioned equation, we suggest quadrature formulas of the vortex loop method with regularization, which provide its approximation on the entire surface when using an unstructured partition. In the problem in question, the derivative of the unknown density of the double-layer potential, as well as the errors of quadrature formulas, has singularities in a neighborhood of the screen edge. We construct a numerical scheme for the integral equation on the basis of the suggested quadrature formulas and prove an estimate for the norm of the inverse matrix of the resulting system of linear equations and the uniform convergence of the numerical solutions to the exact solution of the supersingular integral equation on the grid.     

An approximation of the surfaces areas using the classical Bernstein quadrature formula

In this article, we try to assign a place on the map of the closed Newton–Cotés quadrature formulas to a new approximation formula based on the classical Bernstein polynomials. We create a procedure for a computer implementation that allows us to verify the accuracy of the new approximation formula. In order to get a complete image of this kind of approximation, we compare some well‐known quadrature formulas. Although effective in most situations, there are instances when the composite quadrature formulas cannot be applied, as they use equally‐spaced nodes. We present also an adaptive method that is used to obtain better approximations and to minimize the number of function evaluations. Numerical examples are given to increase the validity of the theoretical aspects.     

Some criteria for maximal abstract monotonicity

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel??s duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators.     

GENUINE-OPTIMAL CIRCULANT PRECONDITIONERS FOR WIENER-HOPF EQUATIONS

1. IntroductionWienerHopf equations are integral equations defined on the haif line:where rr > 0, a(.) C L1(ro and g(.) E L2(at). Here R = (--oo,oo) and ty [0,oo). Inou-r discussions, we assume that a(.) is colljugate symmetric, i.e. a(--t) = a(t). WienerHop f equations arise in a variety of practical aPplicatiolls in mathematics and ellgineering, forinstance, in the linear prediction problems fOr stationary stochastic processes [8, pp.145--146],diffuSion problems and scattering problems […     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

Numerische Quadratur bei Projektionsverfahren

Summary In this paper we investigate the influence of the numerical quadrature in projection methods. In particular we derive conditions for the order of the quadrature formulas in finite element methods under which the order of convergence is not perturbed. It seems that this question has been discussed only for the Ritz method. There is an essential difference between this method on one side and the Galerkin and least squares methods on the other side. The methods using numerical integration are only in the latter case still projection methods. The resulting conditions for the quadrature formulas are often much weaker than those for the Ritz method. Numerical examples using cubic splines and polynomials show that the conditions derived are realistic. These examples also allow the comparison of some projection methods.

     

The modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves

In this article we present and analyze the modified quadrature method for strongly elliptic boundary integral equations of negative order on smooth closed curves. The method employs a modification that extracts the singularity appearing in the kernel or in some of its derivatives. Moreover, the composite trapezoidal rule is used for approximation of the integral. The modified quadrature method is proved to have the maximal rate O(h^−β+1) of convergence for operators of order β < 0 in general, and O(h^−β+2) for a large class of operators appearing in applications. Some numerical experiments confirming our theoretical results are also presented.     

Schur‐convexity and quadrature formulae

This paper aims to contribute to the exploration of quadrature formulae by proving that the error of a quadrature formula has the Schur‐convexity property. The emphasis is on the quadrature formulae with the maximum degree of precision. The Schur‐convexity of the error has an interesting implication – the monotonicity of the error. Namely, it turns out that the absolute value of the error is always smaller on a smaller interval (of the two which share the same midpoint).