多元非理想插值问题研究

项目名称： 多元非理想插值问题研究

项目编号： No.11271156

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 雷娜

作者单位： 吉林大学

项目金额： 50万元

中文摘要： 非理想插值（Birkhoff插值）作为多项式插值的一个重要分类，在逼近论、CAGD、应用密码学、PDE求解等诸多方面有着重要应用。但是与理想插值（Lagrange插值和Hermite插值）已经得到的广泛研究和成熟的理论体系相比，非理想插值的理论和算法因其自身的复杂特性还远未得到完善和进一步发展。已有的结果大都采用传统的分析与逼近的方法，研究特殊的插值问题。项目申请人及其合作者从构造性代数几何的角度重新解读剖析该问题，已经将理想插值问题中的一些关键理论和算法推广到了非理想插值情形，初步建立起了多元非理想插值的符号计算理论基础。本项目将继续采用符号计算与数值计算相结合的方法，从三个方面深入研究一般性的多元非理想插值问题，即①给定插值结点和结点上的微商插值条件（即关联矩阵），构造适定的插值空间；②给定插值空间和关联矩阵，判定插值格式正则性并构造适定结点组；③给定插值系统，寻求高效稳定的求解算法。

中文关键词： 非理想插值；Birkhoff插值；正则性；极小单项基；适定节点组

英文摘要： Non-ideal interpolation (i.e. Birkhoff interpolation) as an important branch of multivariate polynomial interpolation has many significant applications in approximation theory, CAGD, applied cryptography, PDE theory, etc. However, in contrast to the wide study and complete theory of ideal interpolation (Lagrange interpolation and Hermite interpolation), its theory and related algorithms are far from complete due to the difficulties in characterizing its complex behaviors. Current achievements in this field can only solve some problems with special interpolation node structures or/and uniform interpolating conditions. Most of them are based on traditional methods such as analysis or approximation theory. In recent years, the applicant and her collaborators have studied non-ideal interpolation from a constructive algebraic geometric point of view. As a result, they generalized some key theorems and algorithms in ideal interpolation to those in non-ideal situations and established a tentative theory for non-ideal interpolation. In this project, by combining methods in symbolic computation and numerical computation, the applicant will continue the study of non-ideal interpolation mainly on three aspects: 1) given interpolation nodes and corresponding derivative conditions (incidence matrix), construct the proper int

英文关键词： non-ideal interpolation；Birkhoff interpolation；regularity；minimal monomial basis；properly posed set of nodes