Piecewise Linear Interpolation With Nonequidistant Nodes

Pointwise and norm estimates for approximation of continuous function by piecewise linear interpolant with nonequidistant nodes are proved. Special attention is paid to the value of the absolute constants in front of the modulus.     

Piecewise Rational Quadratic Interpolation to Monotonic Data

An explicit representation of a piecewise rational quadraticfunction is developed which produces a monotonic interpolantto given monotonic data. The explicit representation means thatthe piecewise monotonic interpolant is easily constructed andnumerical experiments indicate that the method produces visuallypleasing curves. Furthermore, the use of the method is justifiedby a convergence analysis.     

分段线性插值收敛速度的一种估计

给出了分段线性插值收敛速度的一种估计.     

非线性分数阶常微分方程的分段线性插值多项式方法

通过分段线性插值多项式方法构造了一类含有Hadamard有限部分积分的非线性常微分方程的数值离散格式.在时间方向上, 利用分段线性插值多项式方法对分数阶导数项进行近似, 并通过二阶向后差分格式来离散整数阶导数项.经过详细的证明, 得到了收敛精度为O(τmin{1+α,1+β})的误差估计结果.最后,通过数值算例和理论结果的对比直观地说明了理论分析的正确性.     

强相依数据的函数系数部分线性模型的估计

本文讨论在数据是强相依的情况下函数系数部分线性模型的估计．首先,采用局部线性方法,给出该模型函数项函数的估计;然后,使用两阶段方法给出系数函数的估计．并且讨论了函数项函数估计的渐近正态性,以及系数函数估计的弱相合性和渐近正态性.模拟研究显示,这些估计是较为理想的．     

Compactly Supported, Piecewise Affine Scaling Functions on Triangulations

The methods of using intertwining MRAs to find orthogonal scaling functions have previously been applied to one-dimensional MRAs and are here extended to two-dimensional bases. Two examples are constructed from MRAs consisting of continuous, compactly supported, piecewise affine functions of two variables. The resulting scaling functions can be conveniently restricted to compact domains. November 23, 1997. Date accepted: January 22, 1999.     

一类三角域上的三次样条插值(英文)

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

具有分片线性NCP函数的QP-Free可行域方法

In this paper, a QP-free feasible method with piecewise NCP functions is proposed for nonlinear inequality constrained optimization problems. The new NCP functions are piecewise linear-rational, regular pseudo-smooth and have nice properties. This method is based on the solutions of linear systems of equation reformulation of KKT optimality conditions, by using the piecewise NCP functions. This method is implementable and globally convergent without assuming the strict complementarity condition, the isolatedness of accumulation points. Furthermore, the gradients of active constraints are not requested to be linearly independent. The submatrix which may be obtained by quasi-Newton methods, is not requested to be uniformly positive definite. Preliminary numerical results indicate that this new QP-free method is quite promising.     

Algorithm for Optimal Triangulations in Scattered Data Representation and Implementation

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.     

Fully Piecewise Linear Vector Optimization Problems

Journal of Optimization Theory and Applications - We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed...     

左截断相依数据下条件分位数的双核局部线性估计

本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.     

First Passage Time for Brownian Motion and Piecewise Linear Boundaries

We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.     

Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds

   Abstract. We present algorithms for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse—Smale complexes by canceling pairs of critical points in order of increasing persistence.     

Piecewise Linear Time Series Estimation with GRASP

This paper describes a heuristic to build piecewise linear statistical models with multivariate thresholds, based on a Greedy Randomized Adaptive Search Procedure (GRASP). GRASP is an iterative randomized sampling technique that has been shown to quickly produce good quality solutions for a wide variety of optimization problems. In this paper we describe a GRASP to sequentially split an n-dimensional space in order to build a piecewise linear time series model.     

Relaxation Methods for Strictly Convex Regularizations of Piecewise Linear Programs

We give an algorithm for minimizing the sum of a strictly convex function and a convex piecewise linear function. It extends several dual coordinate ascent methods for large-scale linearly constrained problems that occur in entropy maximization, quadratic programming, and network flows. In particular, it may solve exact penalty versions of such (possibly inconsistent) problems, and subproblems of bundle methods for nondifferentiable optimization. It is simple, can exploit sparsity, and in certain cases is highly parallelizable. Its global convergence is established in the recent framework of B -functions (generalized Bregman functions). Accepted 29 October 1996     

二元线性样条函数插值

二元样条函数插值在计算几何与计算机辅助几何设计中有着重要的作用.本文给出了一种矩形剖分上二元线性样条函数进行Lagrange插值时插值适定结点组所满足的拓扑与几何性质,这种性质依赖于二元线性样条函数所决定的分片线性代数曲线.