Algorithm for constructing range restricted histosplines

This paper is concerned with numerical methods in range restricted histopolation. The proposal is to apply splines on refined grids. The ratios of the added split points are considered to be parameters. In this way, by choosing suitable spline classes, range restricted histosplines can always be constructed if the restrictions are compatible with the given histogram. We offer an algorithm for solving the bivariate problem on a rectangular grid which utilizes univariate results as well as tensor product techniques. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis and implementation of a dual algorithm for constrained optimization

This paper analyzes a constrained optimization algorithm that combines an unconstrained minimization scheme like the conjugate gradient method, an augmented Lagrangian, and multiplier updates to obtain global quadratic convergence. Some of the issues that we focus on are the treatment of rigid constraints that must be satisfied during the iterations and techniques for balancing the error associated with constraint violation with the error associated with optimality. A preconditioner is constructed with the property that the rigid constraints are satisfied while ill-conditioning due to penalty terms is alleviated. Various numerical linear algebra techniques required for the efficient implementation of the algorithm are presented, and convergence behavior is illustrated in a series of numerical experiments.This research was supported by the National Science Foundation Grant DMS-89-03226 and by the U.S. Army Research Office Contract DAA03-89-M-0314.We thank the referees for their many perceptive comments which led to substantial improvements in the presentation of this paper.     

A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems

We introduce a new, one-parametric class of NCP-functions. This class subsumes the Fischer function and reduces to the minimum function in a limiting case of the parameter. This new class of NCP-functions is used in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. We present a detailed investigation of the properties of the equation operator, of the corresponding merit function as well as of a suitable semismooth Newton-type method. Finally, numerical results are presented for this method being applied to a number of test problems.     

Remarks on numerical ranges of operators in spaces with an indefinite metric

The numerical range of an operator on an indefinite inner product space (possibly infinite dimensional) is studied. In particular, operators having bounded numerical ranges are characterized, and the angle points of the numerical range and their connections with eigenvalues are described.

     

Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes

The exit rate from a ‘safe region’ plays an important role in dynamic reliability theory with multivariate random loads. For Gaussian processes the exit rate is simply calculated only for spherical or linear boundaries. However, many smooth boundaries, not of any of these types, are asymptotically spherical in variables of lower dimension, having a greater curvature in the remaining variables. As is shown in this paper, the asymptotic exit rate is then simply expressed as the exit rate from a sphere for a process of the lower dimensions, corrected by an explicit factor.The procedure circumvents the need to calculate complicated exit rate integrals for general boundaries, reducing the problem to a Gaussian probability integral for independent variables.A result of independent interest relates the tail distribution for a sum of a noncentral χ²-variable and a weighted sum of squares of noncentral normal variables, to the tail distribution of the χ²-variable only.     

二次系统(Ⅲ)n=0的极限环问题

本文研究了二次系统(Ⅲ)二次系统; Hopf分支; 极限环.国家自然科学基金(19871041)2004年2月20日2006年2月28日本文研究了二次系统(Ⅲ)二次系统; Hopf分支; 极限环.国家自然科学基金(19871041)2004年2月20日2006年2月28日本文研究了二次系统(Ⅲ)二次系统; Hopf分支; 极限环.国家自然科学基金(19871041)2004年2月20日2006年2月28日本文研究了二次系统(Ⅲ)n=0的极限环问题,利用Hopf分支理论,先考察其产生极限环的参数区域,对此外的参数区域,则运用定性分析的方法,分别给出了无环性的证明,并结合文[2]中的一个重要猜测进行讨沦,完善了谢文[5]的结论。     

Quadratic immersed finite element spaces and their approximation capabilities

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces. Dedicated to the 60th birthday of Charles A. Micchelli Mathematics subject classifications (2000) 65N15, 65N30, 65N50, 65Z05. Yanping Lin: Supported by NSERC. Weiwei Sun: This work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project CityU 1141/01P).     

平方泵噪声驱动的单模激光立方模型及光强定态概率分布

合理地引入了平方泵噪声到单模激光的立方模型中. 得到封闭的光强朗之万方程. 基于作者早期关于非线性外噪声的近似福克-普朗克方程(AFPE)的工作，得到了光强概率分布的AFPE，求出了该AFPE的归一化定态概率分布的精确解析表达式，并根据表达式对泵噪声实部和虚部之间的关联系数λ的对称性，得出了若干关于系统定态统计性质的普遍结论. 关键词： 平方泵噪声 单模激光的立方模型 光强定态概率分布 统计性质重复出现     

A Sequential Quadratically Constrained Quadratic Programming Method of Feasible Directions

In this paper, a sequential quadratically constrained quadratic programming method of feasible directions is proposed for the optimization problems with nonlinear inequality constraints. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving only one subproblem which consist of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the functions of the discussed problems, and such a subproblem can be formulated as a second-order cone programming which can be solved by interior point methods. To overcome the Maratos effect, an efficient higher-order correction direction is obtained by only one explicit computation formula. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without the strict complementarity. Finally, some preliminary numerical results are reported. Project supported by the National Natural Science Foundation (No. 10261001), Guangxi Science Foundation (Nos. 0236001, 064001), and Guangxi University Key Program for Science and Technology Research (No. 2005ZD02) of China.