关于函数方程的若干进展

本文介绍了单实变量的函数方程的若干新进展，包括迭代根、Ｓｃｈｒｏｄｅｒ方程和多项式型迭代方程的结果。基本内容有：Ｉ．引言：迭代与相关问题；Ⅱ．迭代根：存在性；Ⅲ．迭代根：唯一性、可微性和分枝；Ⅳ．多项式型迭代方程。     

求解热传导反问题的一种正则化Newton型迭代法

讨论热传导方程求解系数的一个反问题．把问题归结为一个非线性不适定的算子方程后,考虑该方程的Newton型迭代方法．对线性化后的Newton方程用隐式迭代法求解,关键的一步是引入了一种新的更合理的确定(内)迭代步数的后验准则．对新方法及对照的Tikhonov方法和Bakushiskii方法进行了数值实验,结果显示了新方法具有明显的优越性．     

迭代方程λ_if~i(z)=F(z)局部解析解的存在性

本文在复域中讨论一般多项式函数迭代方程用优函数方法研究方程（＊）的局部解析解的存在性问题．     

Banach空间内一类广义混合隐平衡问题组解的存在性和迭代算法

在Banach空间内引入和研究了一类新的广义混合隐平衡问题组．首先,对广义混合隐平衡问题组引入了Yosida逼近映射概念．利用此概念,考虑了一个广义方程问题组并证明了它与广义混合隐平衡问题组的等价性．其次,应用广义方程问题组,建议和分析了计算广义混合隐平衡问题组的近似解的迭代算法．在相当温和的条件下,证明了由算法生成的迭代序列的强收敛性．这些结果是新的并且统一和推广了这一领域内的某些最近结果．     

解算子与右端数据均有扰动的半正定算子方程的动态系统方法

本文研究了求解算子与右端数据均有扰动的第一类半正定算子方程的动态系统方法．证明了相应的动态系统Cauchy问题的整体解存在且收敛于原算子方程的解．此外,给出了解Cauchy问题的迭代方法并证明了方法的收敛性．     

关于增生算子方程解的带误差的Ishikawa迭代程序

该文在Banach空间中证明了，带误差的Ishikawa迭代序列强收敛到Lipschitz连续的增生算子方程的唯一解．而且，也给Ishikawa迭代序列提供了一般的收敛率估计．利用该结果还推得，带误差的Ishikawa迭代序列也强收敛到Lipschitz连续的强增生算子方程的唯一解．     

分式线性函数的亚纯迭代根

迭代根问题是嵌入流的一个弱问题.关于单调函数的迭代根已有较多结论.但是对非单调函数迭代根的研究却很困难的.分式线性函数是一类实数域上的非单调函数．本文对复平面上分式线性函数的迭代根进行了研究.将分式线性函数的迭代函数方程与一个商空间上的矩阵方程对应,并运用一个求解矩阵根的方法,得到其所有亚纯迭代根的一般公式.并且确定了不同情形下分式线性函数迭代根的准确数目. 作为应用,分别给出了函数$z$和函数$1/z$全部亚纯迭代根.     

无穷区间上一类不连续非线性积分方程的唯一解

在一般Banach空间中研究了一类无穷区间上不连续非线性积分方程的唯一解．在非常弱的条件下证明了非线性积分方程的唯一解可以由迭代序列的一致极限得到，并给出了逼近解的迭代序列的误差估计式，然后应用到无穷区间一阶微分方程的终值问题，本质改进（将紧型条件删去）并推广了一些结果．     

一类Riccati矩阵方程广义自反解的双迭代算法

本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.     

φ-强增生型算子方程的零点逼近

在一般的Banach空间中讨论了φ-强增生算子方程的琴点和φ-强伪压缩映象不动点的迭代逼近问题．     

一维映射迭代根的非单调性及光滑性

迭代根问题是动力系统嵌入流问题的弱问题,是动态插值方法的基础.然而,即使是对一维映射,迭代根的非单调性和全局光滑性都是困难的问题.本文介绍这方面的若干新结果,尤其是关于严格逐段单调连续函数的连续迭代根的存在性和构造,以及迭代根局部光滑与全局光滑的新进展.最后给出多项式迭代根这类既严格逐段单调又具光滑性的迭代根的存在条件及计算方法.     

变系数高维迭代方程的C~1解(英文)

通过构造一个新的结构算子,应用Schauder不动点定理,研究了变系数高维多项式型迭代方程的光滑解.     

Family constraining of iterative algorithms

In constraining iterative processes, the algorithmic operator of the iterative process is pre-multiplied by a constraining operator at each iterative step. This enables the constrained algorithm, besides solving the original problem, also to find a solution that incorporates some prior knowledge about the solution. This approach has been useful in image restoration and other image processing situations when a single constraining operator was used. In the field of image reconstruction from projections a priori information about the original image, such as smoothness or that it belongs to a certain closed convex set, may be used to improve the reconstruction quality. We study here constraining of iterative processes by a family of operators rather than by a single operator.     

Modified Landweber Iteration in Banach Spaces—Convergence and Convergence Rates

We introduce and discuss an iterative method of modified Landweber type for regularization of nonlinear operator equations in Banach spaces. Under smoothness and convexity assumptions on the solution space we present convergence and stability results. Furthermore, we will show that under the so-called approximate source conditions convergence rates may be achieved by a proper a-priori choice of the parameter of the presented algorithm. We will illustrate these theoretical results with a numerical example.     

Estimating the support of multivariate densities under measurement error

We consider the problem of estimating the support of a multivariate density based on contaminated data. We introduce an estimator, which achieves consistency under weak conditions on the target density and its support, respecting the assumption of a known error density. Especially, no smoothness or sharpness assumptions are needed for the target density. Furthermore, we derive an iterative and easily computable modification of our estimation and study its rates of convergence in a special case; a numerical simulation is given.     

DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION

This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian^[7] and by Kulczycki M, Tabor J.^[3] are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.     

Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

Going Off the Grid: Iterative Model Selection for Biclustered Matrix Completion

We consider the problem of performing matrix completion with side information on row-by-row and column-by-column similarities. We build upon recent proposals for matrix estimation with smoothness constraints with respect to row and column graphs. We present a novel iterative procedure for directly minimizing an information criterion to select an appropriate amount of row and column smoothing, namely, to perform model selection. We also discuss how to exploit the special structure of the problem to scale up the estimation and model selection procedure via the Hutchinson estimator, combined with a stochastic Quasi-Newton approach. Supplementary material for this article is available online.     

The template‐based procedure of biorthogonal ternary wavelets for curve multiresolution processing

Curve multiresolution processing techniques have been widely discussed in the study of subdivision schemes and many applications, such as surface progressive transmission and compression. The ternary subdivision scheme is the more appealing one because it can possess the symmetry, smaller topological support, and certain smoothness, simultaneously. So biorthogonal ternary wavelets are discussed in this paper, in which refinable functions are designed for cure and surface multiresolution processing of ternary subdivision schemes. Moreover, by the help of lifting techniques, the template‐based procedure is established for constructing ternary refinable systems with certain symmetry, and it also gives a clear geometric templates of corresponding multiresolution algorithms by several iterative steps. Some examples with certain smoothness are constructed.     

Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization

Crossed symmetric solutions of nonlinear boundary value dynamic problems play an important role in many applications, in particular in adaptive algorithm designs. This article is devoted to the continuation of our investigation on second-order nonlinear companion dynamic boundary value problems on time scales. Monotonically convergent upper and lower solutions of the problems and their quasilinear approximations are investigated. It is shown that, under proper smoothness constraints, the iterative sequences constructed not only converge to the analytic solutions of the desired companion problems monotonically, but also preserve important crossed symmetry properties. The quasilinearization offers an efficient way in the solution approximation. Computational examples are given to illustrate our results.