一类融合逼近和插值的曲线细分

采用生成多项式为主的方法对一类融合逼近和插值三重细分格式的支撑区间、多项式生成、连续性、多项式再生及分形性质进行了分析,给出并证明了极限曲线C^k连续的充分条件.通过对融合型细分规则中参数变量的适当选择来实现对极限曲线的形状调整,从而衍生出具有良好性质的新格式,并将这类新格式与现有格式进行比较.数值实例表明这类新格式生成的极限曲线具有较好的保形性.     

五点二重融合型细分法

提出了一种新的细分算法——五点二重融合型细分法.利用生成多项式对该细分法的一致收敛性和C~k连续性进行了分析,通过对融合型细分法中参数的不同取值,可以分别生成C~1～C~6连续的极限曲线.数值实例表明,与现有一些格式相比,细分算法生成的极限曲线不仅可以保持较高光滑性,并且更接近初始控制多边形.     

一类保凸的细分格式

提出了一类新的具有高阶连续性和保凸性的5-点二重细分格式.可以证明在参数的某些取值范围内,极限曲线是$C^k~(k=0,1,\ldots,7)$连续的.本文还给出了极限曲线保凸时的参数的取值范围.数值例子表明该格式是灵活有效的.     

一类具有优良性质的五点二重逼近细分格式（英文）

提出了一类带两个形状参数的五点二重逼近细分格式.这类格式具有一些优良的性质:高阶连续性、可调性和多项式再生性质.对于参数的某些取值范围极限曲线可以达到C~k(k=0,1,…,7)连续,分析了一类特殊情况的多项式再生性质.实例表明该格式的有效性和灵活性.     

六点Binary逼近细分法

提出了一种新的细分算法——六点Binary逼近细分法.利用生成多项式等方法对细分法的一致收敛性和Ck连续性进行了分析,通过对细分法中张力参数μ的不同取值,极限曲线可达到C~0~C~7连续.特别是当μ=11/1024时,极限曲线可达到C9连续.数值算例表明,该方法是合理有效的.     

插值细分曲线有理参数点的精确求值

本文提出了求值插值细分曲线上任意有理参数的算法.通过构造与细分格式相关的矩阵,m进制分解给定有理数以及特征分解循环节对应算子乘积,计算得到控制顶点权值,实现对称型静态均匀插值细分曲线的求值.本文给出了四点细分和四点Ternary细分曲线的求值实例.算法可以推广到求值其他非多项式细分格式中.     

双参数六点细分法

提出了一类包含两个形状参数的双参数六点细分法,可以构造光滑插值曲线和光滑逼近曲线,并且可以通过对两个参数取值的调整使得曲线达到一致收敛,C1或C2.讨论了形状参数对细分法的收敛性及连续性的影响,给出了细分法一致收敛、C1连续、C2连续的充分条件,并给出了一些数值算例.     

曲线一致细分算法中极限曲线的几何性质

曲线的一致细分算法是计算机图形与图象处理中的一处重要的快速生成曲线的剖分算法。本文给出了曲线一致细分算法中极限曲线的三个几何性质。     

几何造型的有理矩阵细分方法

Ｍｉｃｃｈｅｌｌｉ,Ｐｒａｕｔｚｓｃｈ给出了一类生成曲线的细分法－矩阵细分方程,但该方法仅能生成多项式类型的曲线,为了弥补其不足,本文提出了有理矩阵细分方法,并证明了其生成曲线的优良性质,例如凸包性,几何不变性,变差缩减性等。     

具有多项式衰减面具的向量细分格式

具有多项式衰减面具的向量细分方程在刻画小波Riesz基和双正交小波等方面有着重要作用.本文主要研究这类方程解的性质.向量的细分方程具有形式：Ф=∑α∈Zsa（α）（2·-α）,其中Ф=（Ф1,...,Фr）T是定义在Rs上的向量函数,a：=（a（α））α∈Zs是一个具有多项式衰减的r×r矩阵序列称为面具.关于面具a定义一个作用在（Lp（Rs））r上的线性算子Qa,Qaf：=∑α∈Zsa（α）f（2·α）.迭代格式（Qanf）n=1,2,...称为向量细分格式或向量细分算法.本文证明如果具有多项式衰减面具的向量细分格式在（L2（Rs））r中收敛,那么其收敛的极限函数将自动具有多项式衰减.另外,给出了当迭代的初始函数满足一定的条件时的向量细分格式的收敛阶.     

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C ¹ smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots—that are not considered by the authors of the corresponding spline basis—thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.     

A family of optimal fourth-order methods for multiple roots of nonlinear equations

Newton-Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton-type method at the first step. In this paper, we present a new two-step optimal fourth-order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.     

CHARACTERIZATION OF COVERGENT SUBDIVISION SCHEMES

Subdivision schemes provide important techniques for the fast generationof curves and surfaces.A recusive refinement of a given control polygonwill lead in the limit to a desired visually smooth object.These methodsplay also an important role in wavelet analysis.In this paper,we use arather simple way to characterize the convergence of subdivision schemesfor multivariate cases.The results will be used to investigate the regularityof the solutions for dilation equations.     

On Some Classical Methods to Improve Algebraic Curves and Surfaces

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

一类新的细分曲线方法

Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements based on initial control polygon or grid.Usually the curve refinements is the basis of the corresponding surface rules. In this paper we analyze previous subdivision scheme according to theories about convergence of N.Dyn and M.F Hassan. In terms of binary and ternary subdivision schemes general construction about curve‘s refinements are studied.Two approximating curve subdivision schemes with neighboring four control points are derived,the generating limit curves can both reach the smoothness of C^1 over the initial polygon using the two schemes and the tolerances of them are given according to the method of [7].     

Auswertungsverfahren für Polynome in mehreren Variablen

A general class of evaluation schemes for polynomials in one or several variables is discussed. By the same concept, error bounds are obtained for various methods, for instance Horner's scheme and Clenshaw's method, which are strict in some cases with a loss of a factor logn at best. For multivariable polynomials, a new family of evaluation schemes is suggested which generalizes a modification of Clenshaw's method and is therefore expected to have a favorable stability behavior with respect to round-off.     

奇异积分算子q-变差的定量最优加权估计

本文将定量最优A_p权理论推广到联系于ω-Calderón-Zygmund算子的q-变差情形.这些结果利用了Lerner最新给出的稀疏控制方法来控制q-变差,和Hyt?nen等关于q-变差的最优加权成果相比,本文涉及的ω仅需满足Dini条件,并且其截断是非光滑的.     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.