Sparse approximate solution of fitting surface to scattered points by MLASSO model

The goal of this paper is to achieve a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant(PSI) space and the l_1 norm minimization. In order to obtain different sparsity of the approximation solution, the problem is represented as a multilevel LASSO(MLASSO)model with different regularization parameters. The MLASSO model can be solved efficiently by the alternating direction method of multipliers. Numerical experiments indicate that compared to the AGLASSO model and the basic MBA algorithm, the MLASSO model can provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution. Moreover, the solution by the MLASSO model can reflect the regions of the underlying surface where high gradients occur.  

基于离散点和法矢的曲线重构算法

本文提出了一个基于离散点及法矢的B样条曲线重构算法.本方法在以下三个步骤做出了改进:基于法矢条件的离散点参数化,基于由法矢信息选取主导点的B样条节点向量确定方法,以及在拟合模型中平衡数据点误差及法矢向量误差的参数选取方法.由此,我们将B样条曲线拟合问题转化为三个相应子问题,并且能够自适应地得到B样条曲线.对比仅由数据点选取主导点的传统拟合方法,本文方法对一些包含较高噪音的数据集仍然能较好地保持原曲线的几何形状.  

基于层次T网格的分片孔斯曲面重构

本文提出一种基于任意层次T网格的多项式(PHT)样条空间$S(3,3,1,1,T)$的一个新的曲面重构算法.该算法由分片插值于层次T网格上每个小矩形单元对应4个顶点的16个参数的孔斯曲面形式给出.对于一个给定的T网格和相应基点处的几何信息(函数值,两个一阶偏导数和混合导数值),可得到与$S(3,3,1,1,T)$的PHT样条曲面相同的结果,且曲面表达形式更简单,同时,在离散数据点的曲面拟合中,我们给出了自适应的曲面加细算法.数值算例显示,该自适应算法能够有效的拟合离散数据点.  

基于MFS的PDE-约束优化法求解热传导逆问题

在本文中,我们给出了一种有效的无网格方法来求解逆热传导问题,含有Neumann边界条件情形.所得到的PDE-约束优化法是一种在空间与时间域上的全局近似方法,其中将控制方程的基本解作为基函数.由于初始测量数据包含有噪声误差,则所得线性方程组的系数矩阵通常是病态的,文中利用广义交叉验证(GCV)的Tikhonov正则化方法来获得更加稳定的数值解.通过数值结果表明,本文给出的方法是精确、有效、鲁棒的.  

Reconstruction of the Linear Ordinary Differential System Based on Discrete Points

In this paper, we discuss an inverse problem, i.e., the reconstruction of a linear differential dynamic system from the given discrete data of the solution. We propose a model and a corresponding algorithm to recover the coefficient matrix of the differential system based on the normal vectors from the given discrete points, in order to avoid the problem of parameterization in curve fitting and approximation. We also give some theoretical analysis on our algorithm. When the data points are taken from the solution curve and the set composed of these data points is not degenerate, the coefficient matrix $A$ reconstructed by our algorithm is unique from the given discrete and noisefree data. We discuss the error bounds for the approximate coefficient matrix and the solution which are reconstructed by our algorithm. Numerical examples demonstrate the effectiveness of the algorithm.  

截断分层B样条基的Lp稳定性

截断分层B样条基是自适应曲线曲面拟合的重要的工具,它兼顾了宏观形状的构造和局部细节的控制.这种基底的$L_\infty$模的强稳定性在文献[20]中被讨论,但这种基底的$L_p(1\le p\le\infty)$模的稳定性并不清晰.在本文中我们讨论截断分层B样条基的$L_p$稳定性,既然基底的$L_p$稳定性对于形状构造十分重要.我们证得截断分层样条基是弱$L_p$稳定基,即基底稳定性中的系数依赖于分层空间的层数.