高维空间径向基函数拟插值算子构造方法及其应用

项目名称： 高维空间径向基函数拟插值算子构造方法及其应用

项目编号： No.11301252

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 姜自武

作者单位： 临沂大学

项目金额： 22万元

中文摘要： 本项目主要研究高维空间上径向基函数拟插值算子的构造方法及其在数据拟合、曲面造型和偏微分方程数值解中的应用。径向基函数方法作为一种普遍适用的无网格方法，是目前科学和工程计算领域研究的热点，相关理论研究已经获得长足发展。径向基函数拟插值方法由于能够避免求解大规模线性方程组，成为当今径向基函数方法的研究热点之一。申请人已经构造了一种一维空间上高精度径向基函数拟插值算子，并将其应用于数据拟合和偏微分方程数值解，取得了良好的效果。目前针对高维空间上径向基函数拟插值算子的研究还很少。在已有工作的基础上，本项目拟深入探讨如下问题，首先对高维空间数据点进行局部预处理，构造径向基函数拟插值算子；其次根据再生多项式的最高次数，研究所构造的拟插值算子的逼近精度；最后将所构造的拟插值算子有效应用于数据拟合、曲面造型和数值求解偏微分方程。

中文关键词： 径向基函数；拟插值；偏微分方程数值解；循环矩阵；四元数矩阵

英文摘要： The project is mainly focused on the construction of quasi interpolation operator with radial basis function in high dimensional space and its application in data fitting, surface modeling and solving numerical solution of partial differential equations. Radial basis function as a generally applicable meshless method is currently a hotspot in the field of computer science and engineering, and much headway of its theory has been made. Quasi interpolation method of radial basis function can avoid solving large-scale linear equations and become one of the hot spots in the research of radial basis function method. The applicant has constructed a high accuracy quasi interpolation operator, and applied it to data fitting and solving numercial solution of partial differential equations, and achieved good results. At present, research on the quasi interpolation of radial basis function in high dimensional space is rarely. Motivated by preliminary works, members of this team will give in-deep study of the following problem. Firstly, according to local preconditioning on high dimensional data points, constructing quasi interpolation operator of radial basis function; Secondly, by using the maximum degree of polynomials regenerated by this opterator, we study the approximation accuracy of the presented quasi interpolation

英文关键词： Radial basis function；Quasi-interpolation；Numerical method of PDE；Circulant Matrix；Quaternion