基于周期数据的广义保形拟插值的理论及其应用

项目名称： 基于周期数据的广义保形拟插值的理论及其应用

项目编号： No.11501006

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 高文武

作者单位： 安徽大学

项目金额： 18万元

中文摘要： 拟插值被广泛地应用于处理各种逼近问题,它不需要求解线性方程组就可以直接给出逼近函数,具有计算量小、计算稳定性好等优点.而对于许多逼近问题,被逼近对象往往具有某种周期性.然而,经典的拟插值在处理这类问题时面临两个困难:1.需要在边界进行周期延拓(这不仅需要边界条件,而且还会在边界处出现高阶不连续点,产生延拓的痕迹);2.保形性(保单调性/凸性)的性质失去意义(周期函数在单个周期内不再是单调/凸函数).因此,需要研究基于周期数据的拟插值理论及其应用.利用数值逼近、计算几何的理论和方法,融合 multiquadric 拟插值和三角样条拟插值的构造思想,本项目将研究该课题.主要包括:探讨基于周期数据的拟插值构造理论和基本方法,推导拟插值的性质如对函数及其高阶导数的逼近阶、广义保形性、稳定性和形状参数的最佳选择准则,研究拟插值在微分方程数值解、数值微分、计算机辅助几何设计等领域的应用.

中文关键词： 拟插值；逼近阶；周期性；广义差商；广义保形性

英文摘要： Quasi-interpolation has been widely used in dealing with approximation problems. It yields a solution directly without the need to solve any linear systems, and thus it possesses the properties of taking small amount of computation and fair stability of computation.For many approximation problems,the approximand usually possesses some kind of periodicity.However,these approximation problems,most classical quasi-interpolation faces two difficulties:1.they need to do periodic extensions on the boundaries,which not only requires some boundary conditions,but also yields high-order discontinuous points at the boundaries and thus gives extension trails;2.The shape-preserving(monotonicity/convexity-preserving)property is not valid since periodic functions can not be monotonic/convex in a period any more.Thus,it needs to study theory and application of quasi-interpolation for periodic data.Coupled together the construction ideas of MQ quasi-interpolation and trigonometric spline quasi-interpolation,with the help of the theories and methods of numerical approximation,computational geometry,the project will study such a topic.This project mainly consists of studying theories and basic methods of constructing quasi-interpolation for periodic data,deriving some properties of the quasi-interpolant(approximation orders for a function and its high-order derivatives,generalized shape-preserving properties,stability and optimal choices of the shape parameter),exploring its applications in numerical solution of differential equations,numerical differention,computer aided geometric design and so on.

英文关键词： Quasi-interpoation;Approximation order;Periodicity;Generalized divided difference;Generalized shape-preserving property