Approximation processes for resolvent operators

We introduce general sequences of linear operators obtained from classical approximation processes which are useful in the approximation of the resolvent operators of the generators of suitable C ₀-semigroups. The main aim is the representation of the resolvent operators in terms of classical approximation operators. Work performed under the auspices of PRIN 2006–07 “Kolmogorov equations” (coordinator G. Da Prato)     

Polynomial expansions of boolean functions in images of nonhomogeneous operators

To represent the functions in terms of which expansions are constructed, the so-called dp-, pt -, and dt -operators are used. The existence of polynomial expansions of Boolean functions in functions obtained by applying heterogeneous operators of the above-mentioned types to a nondegenerate Boolean function is proved. Some methods of finding coefficients of these expansions are also considered. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 40–55, May–June, 2000.     

A method for the approximation of distributions

Le but de cet article est l‘étude d‘une méthode d‘approximation de distributions par des combinaisons linéaires de masses de Dirac. La valeur des coefficients de cette combinaison lineaire est dérminée explidtement, en utilisant des suites de fonctions tendant vers la mesure de Dirac. Nous pr£sentons d‘abord le principe de la méthode puis quelques résultats théoriques de convergence. Enfin, nous donnons des résultats numériques d‘approximation de distributions et nous appliquons la méthode à la résolution numerique d'equations integrates singulieres.

The object of this article is to study a method of approximation of distributions by linear combinations of Dirac masses. The value of the coefficients of this linear combination is explicitly determined by using sequences of functions tending towards the Dirac measure. First, we present the principle of the method, then some theoretical results of convergence. Finally, we give numerical results of approximation of distributions and we apply the method to the numerical solution of singular integral equations.     

A family of simultaneous methods for the determination of polynomial complex zeros

Using the development of a rational function by elementary fractions, a family of methods for the simultaneous determination of polynomial complex zeros is derived. All the methods of the family are cubically convergent for simple zeros. The known simultaneous procedures of the third order are included. The presented class of iteration functions is suitable for the parallel inclusion of polynomial complex zeros by circular regions. The family of methods, defined in complex circular arithmetic, gives a new interval method with cubic convergence. Numerical example is given.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

Convergence properties of eigenvalues of space discretized diffusion equations

A theorem is presented in this paper to establish the convergence of eigenvalues of space discretized Diffusion equation. The computational results confirm this.     

Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints

The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large-scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this paper, we give a partial answer on this problem. Specially, under the assumption that the associated function satisfies the Kurdyka–?ojasiewicz inequality, we prove that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions. Under some further conditions on the problem's data, we also analyse the convergence rate of the algorithm.     

An estimate on the convergence of MKZ–Bézier operators

The pointwise approximation properties of the MKZ–Bézier operators M_n,α(f,x) for α≥1 have been studied in [X.M. Zeng, Rates of approximation of bounded variation functions by two generalized Meyer–König–Zeller type operators, Comput. Math. Appl. 39 (2000) 1–13]. The aim of this paper is to study the pointwise approximation of the operators M_n,α(f,x) for the other case 0<α<1. By means of some new estimate techniques and a result of Guo and Qi [S. Guo, Q. Qi, The moments for the Meyer–König and Zeller operators, Appl. Math. Lett. 20 (2007) 719–722], we establish an estimate formula on the rate of convergence of the operators M_n,α(f,x) for the case 0<α<1.