近似最优径向基函数插值的理论与算法研究

项目名称： 近似最优径向基函数插值的理论与算法研究

项目编号： No.11301045

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： 方芩

作者单位： 大连大学

项目金额： 22万元

中文摘要： 径向基函数插值的逼近质量依赖于插值结点的分布及潜在的被插值函数。现有的结点选取算法大多不考虑被插值函数，且缺乏最优性的严格证明。非线性逼近的基本思想是用来逼近的函数不来自于一个固定的线性空间，且可以依赖于被逼近函数。这促使我们试图利用非线性逼近中的方法与工具，将插值结点的选取和被插值函数结合起来，研究近似最优径向基函数插值及近似最优插值结点的理论与算法。主要研究如下三个问题：（1）定义关联于结点集的子空间与关联于最优m项逼近的子空间的距离，以便将结点选取问题转化为空间逼近问题；（2）利用贪婪算法求解涉及关联于结点集的子空间与关联于最优m项逼近的子空间的空间逼近问题，并研究算法的收敛速率；（3）研究由贪婪算法所生成插值结点集的近似最优性。该项目的成功实施，将对机器学习、曲面重构以及无网格微分方程数值解等领域中径向基函数插值逼近性能的改善产生积极影响。

中文关键词： 径向基函数；非线性逼近；近似最优；插值；

英文摘要： The approximation quality of interpolation using radial basis functions (RBF) depends on the distribution of interpolating knots, and the underlying interpolated function. The existing algorithms for selecting interpolating knots are usually independent of interpolated functions, and have no rigorous argument for optimization. The basic idea behind nonlinear approximation is that the functions used in the approximation do not come from a fixed linear space but are allowed to depend on the approximated function. We are motivated to develop theories and algorithms for near optimal RBF interpolation, and near optimal interpolating knots by using the nonlinear approximation theory and by exploring relationship between interpolating knots and the interpolated function. The research mainly consists of three topics: (1)proposing an appropriate distance between the interpolating-knot-related subspace and the optimal-m-term-approximant-related subspace, so that one can reduce the seclection of interpolating knots to a space approximation; (2)developing a greedy algorithm for solving the space approximation involving the interpolating-knot-related subspace and the optimal-m-term- approximant-related subspace, and studying the convergence rate of the algorithm; (3) studying the near optimization of interpolating knots gen

英文关键词： Radial basis function；Nonlinear approximation；Near optimal；Interpolation；