相对φ强伪压缩算子不动点的迭代逼近

在没有任何有界以及迭代参数列不必收敛于零等条件下,使用新的分析方法建立了相对φ强伪压缩算子不动点与相对φ强增生算子方程解具误差Ishikawa迭代序列的强收敛定理,推广和改进了有关文献中的相应结果,而且还给出了收敛率的估计式.     

Banach空间中多值Φ-增生和多值Φ-伪压缩映象的收敛定理

在任意Banach空间中引入比重要的多值-强增生映象和多值-强伪压缩映象更一般的多值Φ-增生映象和多值Φ-伪压缩映象,研究多值Φ-增生映象方程的解和多值Φ-伪压缩映象不动点的具随机误差的Ishikawa迭代逼近问题.这些结论推广和改进了最新文献中相应的结果.     

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results. Received November 29, 1996 / Revised version received April 25, 1997     

A New Third-Order Iterative Processfor Solving Nonlinear Equations

 In this paper, we build up a modification of the Midpoint method, reducing its operational cost without losing its cubical convergence. Then we obtain a semilocal convergence result for this new iterative process and by means of several examples we compare it with other iterative processes. (Received 11 April 2000; in final form 27 March 2001)     

An iterative multi level algorithm for solving nonlinear ill–posed problems

Summary. The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations, when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization. An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The numerical performance of this multi level algorithm is compared with Landweber's iteration. Received October 21, 1996 / Revised version received July 28, 1997     

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones. Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999     

On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals

In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Gröbner basis.     

Kronecker type theorems,normality and continuity of the multivariate Padé operator

Summary. For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function being rational, is well-known. In [Lubi88] Lubinsky proved that if is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84]. Received December 19, 1994     

On Optimal Constants for Best Two-Dimensional Simultaneous Diophantine Approximations

 The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two. (Received 29 April 1998; in revised form 23 November 1998)     

A convergence analysis of the Landweber iteration for nonlinear ill-posed problems

Summary. In this paper we prove that the Landweber iteration is a stable method for solving nonlinear ill-posed problems. For perturbed data with noise level we propose a stopping rule that yields the convergence rate ) under appropriate conditions. We illustrate these conditions for a few examples. Received February 15, 1993 / Revised version received August 2, 1994     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

Banach空间中具增生映象的变分包含解的存在性和逼近问题

研究了Banach空间中一类增生型变分包含解的存在性及其迭代逼近问题。所得结果改进和推广了一些人的最新成果。     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998     

Rational approximation to orthogonal bases and of the solutions of elliptic equations

It is shown that given , an orthogonal basis of can be approximated by an orthogonal basis , where has integral and have rational components, such that the angle between and is at most and the length , . This improves the length of the integral approximation due to Schmidt (1995). As an application, we improve a theorem of Kocan (1995) about the minimal size of grids in the solutions of elliptic equations. Our result fits the need in Kuo and Trudinger (1990). Received January 27, 1997 / Revised version received April 1, 1998     

Norm inequalities related to operator monotone functions

Let A,B be positive semidefinite matrices and any unitarily invariant norm on the space of matrices. We show for any non-negative operator monotone function f(t) on , and for non-negative increasing function g(t) on with g(0) = 0 and , whose inverse function is operator monotone. Received: 1 February 1999     

Stochastic heat equation with random coefficients

We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from It?'s formula for this anticipating stochastic integral. Received: 21 November 1997 / Revised version: 20 July 1998     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail. (Received 26 April 2000; in final form 17 November 2000)