WEAK REGULARIZATION FOR A CLASS OF ILL-POSED CAUCHY PROBLEMS

This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibility method and regularized semigroups. Finally, an example is given.     

Some approximation properties by a class of bivariate operators

Starting with the well‐ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBS‐type operator are compared.     

A discrepancy principle for equations with monotone continuous operators

A discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated. The existence and uniqueness of the corresponding regularization parameter a(δ) are proved. Convergence of the solution obtained by the discrepancy principle is justified. The results are obtained under natural assumptions on the nonlinear operator.     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

The finite rank perturbations of the product of Hankel and Toeplitz operators

In this paper we completely characterize when the product of a Hankel operator and a Toeplitz operator on the Hardy space is a finite rank perturbation of a Hankel operator, and when the commutator of a Hankel operator and a Toeplitz operators has finite rank.     

一类非线性不适定问题的Tikhonov正则化

本文在B anach空间中,研究一类非线性不适定问题的正则化,所研究的算子是多值的,且是近拟的,假定原始问题是可解的,利用带双参数的T ikhonov正则化方法构造出逼近步骤.     

A Generalization of Tyuriemskih's Lethargy Theorem and Some Applications

Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator actually converges arbitrarily slowly. Then this generalization is used to answer affirmatively a 2010 conjecture of ours as well as establishing that all of the classical operators of Bernstein, Hermite-Fejer, Landau, Fejer, and Jackson converge arbitrarily slowly to the identity operator (and not just almost arbitrarily slowly as we established in 2010).     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

A Way of Estimating the Convergence Rate of the Fourier Method for PDE of Hyperbolic Type

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.     

A—光滑正则化算子

本文研究了紧算子方程的Moore-Penrose广义解的逼近,引进了A-导数的概念和对应的A-光滑正则化算子.这个双参数的A-光滑正则化算子族有明显的变分意义,并且包含正则化算子作为它的特殊情形,以(修正的)截断奇异值分解方法作为它的极限情形.这些正则化算子的性质表明它们有广阔的实际应用可能性.     

(NCI)算子与(NFI)算子

引入两类与不变子空间密切相关的算子,即(NCI)算子与(NFI)算子.考虑这两类算子的性质,例如存在性,相似不变性,以及与强不可约算子的关系等.并在可分Banach空间上讨论谱为单点集{0}的算子能否小紧摄动成为(NFI)算子.     

Numerical pseudodifferential operator and Fourier regularization

The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.     

The effects of a new selection operator on the performance of a genetic algorithm

In this study, the new back controlled (BCSO) selection operator for genetic algorithm (GA) is presented. In the first stage, six existing operators and the BCSO were applied to the traveling salesman and space truss benchmark problems. An analysis of the results of the benchmark problems was made for each of the seven selection operators. In the second stage, the BCSO, the Roulette Wheel, dominant selection the hybrid II and operators were used to determine the fittest beam section has been used. In this stage, the cost of beam for a 18.3 m span developed by Mid-Atlantic States Pre-stressed Concrete Committee for Economic Fabrication and using a genetic algorithm the cost of a beam using BCSO were compared. It was determined that the beam produced using the proposed BCSO selection operator would be lower than those obtained with the existing selection operators. Also the determined beam cost is compares favourably with the average cost of a beam sold by the precast industry in the USA.     

An iterative scheme for solving nonlinear equations with monotone operators

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified. AMS subject classification (2000)  47J05, 47J06, 47J35, 65R30     

On a Monotone Ill-posed Problem

A class of a posteriori parameter choice strategies for the operator version of Tiknonov regularization （including variants of Morozov＇s and Arcangeli＇s methods） is proposed and used in investigating the rate of convergence of the regularized solution for ill-posed nonlinear equation involving a monotone operator in Banach space.     

The Edge Algebra Structure of Boundary Value Problems

Boundary value problems for pseudodifferential operators (with orwithout the transmission property) are characterised as a substructureof the edge pseudodifferential calculus with constant discreteasymptotics. The boundary in this case is the edge and the inner normalthe model cone of local wedges. Elliptic boundary value problems fornoninteger powers of the Laplace symbol belong to the examples as wellas problems for the identity operator in the interior with a prescribednumber of trace and potential conditions. Transmission operators arecharacterised as smoothing Mellin and Green operators with meromorphicsymbols.     

复合算子和一类解析Toeplitz算子的约化子空间

本文利用复合算子完全刻画了符号为两个Ｂｌａｓｃｈｋｅ因子乘积的解析Ｔｏｅｐｌｉｔｚ算子的约化子空间。     

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S-spectrum. This allow us to obtain conditions when a non-selfadjoint operator admits a triangular representation.     

A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy

In this paper, we propose variants of Forward-Backward splitting method for finding a zero of the sum of two operators. A classical modification of Forward-Backward method was proposed by Tseng, which is known to converge when the forward and the backward operators are monotone and with Lipschitz continuity of the forward operator. The conceptual algorithm proposed here improves Tseng’s method in some instances. The first and main part of our approach, contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. During the iteration process, the search performs only one calculation of the forward-backward operator in each tentative of the step. This achieves a considerable computational saving when the forward-backward operator is computationally expensive. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without Lipschitz continuity assumption on the forward operator.     

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.