理想插值的误差公式与离散化问题研究

发布时间：2018-02-11 03:04

本文关键词： 理想插值理想投影算子 Gr(?)bner基误差公式离散化　出处：《吉林大学》2016年博士论文　论文类型：学位论文

【摘要】：理想插值最早由数学家Birkhoff提出,用来研究一般的多元多项式插值问题.理想插值是一种被插函数为多项式的线性插值格式,其可看成是经典的一元Lagrange插值与Hermite插值在多元情形下的推广.具体来说,理想插值由理想投影算子确定.理想投影算子是多项式空间到自身的线性幂等算子,其核恰为一多项式理想.在理想插值中,理想投影算子的像空间为插值空间,理想投影算子的对偶的像空间为插值条件泛函空间.插值条件泛函空间由一组插值节点,以及每个节点上相应的赋值泛函与由有限维微分闭子空间所定义的微分算子的复合构成.微分闭子空间是由多项式构成的线性空间,并且其对求导运算是封闭的.由于“微分闭”的概念是对一元Hermite插值条件中“连续阶”导数要求的推广,所以理想插值包含了经典的Lagrange插值与Hermite插值,其中Lagrange插值对应的理想投影算子称为Lagrange投影算子.2005年,de Boor在他的理想插值综述中提到下列问题,其一：理想投影算子是否具有统一的误差结构表达式；其二：哪些理想投影算子具有“好”误差公式；其三：若一理想投影算子为Hermite投影算子,如何计算逼近它的Lagrange投影算子列.到目前为止,这些问题仍然是理想插值中的研究热点.为简便起见,我们称前两个问题为理想插值的误差公式问题,称最后一个问题为理想插值的离散化问题.本文将利用代数几何的理论知识研究上述问题,并给出一些理论结果.主要工作如下：1.给出了理想投影算子统一的误差结构表达式.一元理想投影算子的误差的结构形式简单优美.为将其推广到多元情形,de Boor提出了理想投影算子的“好”误差公式的概念.“好”误差公式是一种误差结构表达式,具体说,是指存在齐次多项式Hj和线性算子q使得插值误差可以表示为f-Pf＝ ∑j=1mCj(Hj(D)f)hj且满足正交条件Hj(D)hk=δj,k,其中f为被插多项式函数,P为理想投影算子,马(D)为微分算子,{h1,...,hm}为理想kerP的理想基de Boor曾猜测所有理想投影算子都具有“好”误差公式,但随后Shekhtman给出了一个二元情形下的反例,并断言大多数理想投影算子都不具有“好”误差公式.我们研究了理想投影算子的误差公式的代数结构,在“好”误差公式的基础上,提出了“一般”型误差公式的概念.然后利用理想的约化理论,证明了所有理想投影算子的核空间的字典序下的约化Grobner基都支撑“一般”型误差公式.最后利用B样条理论,给出了Shekhtman反例的“一般”型误差公式的具体表达式.2.给出了一类Lagrange投影算子的“好”误差公式的具体表达式.前面我们提到不是所有理想投影算子都具有“好”误差公式.到现在为止,人们对“好”误差公式的存在性的研究取得了一定进展Shekhtman证明了特殊几何分布节点上的理想投影算子具有“好”误差公式,李U喼っ髁司叻篏robner基的理想投影算子有“好”误差公式,de Boor给出了具张量积节点和满足GC条件节点的Lagrange投影算子的“好”误差公式的具体表达式.受这些工作的启发,我们研究了一类特殊理想投影算子的误差公式.针对Cartesian点集上的Lagrange投影算子,首先利用差商算法,给出插值余项.然后将插值余项整理成差商形式的“好”误差公式.最后利用差商与样条积分的关系,给出了“好”误差公式的具体表达式.3.研究了一类二元Hermite投影算子的离散化问题.当人们推广一个概念时,一般会保留原有的结构属性.在一元情形下,Hermite插值是Lagrange插值的极限形式.这一事实启发de Boor定义Hermite投影算子为Lagrange投影算子的极限.虽然一元理想投影算子都是Hermite投影算子,并且这个结论在某些多元情形下也成立,但已有例子表明存在非Hermite的多元理想投影算子.所以判断一个理想投影算子是否为Hermite投影算子,以及如何计算逼近Hermite投影算子的Lagrange投影算子列是人们十分关心的问题.围绕这个问题(理想插值的离散化问题),de Boor和Shekhtman证明了二元理想投影算子都是Hermite投影算子,并给出了一种计算其相应Lagrange投影算子列的方法.但是方法本身复杂度高,不易于实现.我们研究了一类特殊的二元Hermite投影算子,其插值条件泛函为δξοΡ(n)(D),Ρ(n):=Fn[x,y](?)spanF{pn},其中δζ为ζ点处的赋值泛函,D为微分算符,Fn[x,y]为次数小于n的二元多项式集合,pn为一任意的二元n次多项式.针对此类Hermite投影算子,我们给出了一种计算Lagrange投影算子列的方法,该方法简单有效并且几乎不用任何计算代价.
[Abstract]:The ideal interpolation was first proposed by mathematician Birkhoff, multivariate polynomial interpolation is used to study the problem in general. The ideal interpolation is an interpolating function for linear interpolation polynomial, which can be regarded as a generalization of the classical Lagrange interpolation and Hermite interpolation in the multivariate case. Specifically, the ideal interpolation is determined by the ideal projection operator. The ideal projection operator is idempotent operator's linear polynomial space, the kernel is a polynomial ideal. The ideal interpolation, the ideal projection operator like space interpolation, dual ideal projection operator like space interpolation conditions. The functional space interpolation functional space consists of a set of interpolation nodes, and the composition of each the corresponding node assignment by finite dimensional differential functionals and closed subspaces defined by differential operators. Differential closed subspace is constructed by polynomials of the linear space, And it is closed to the derivative operations. Due to the "differential closed" concept of Hermite interpolating condition "continuous order derivative" requirements of the promotion, so the ideal interpolation contains the classical Lagrange interpolation and Hermite interpolation, which corresponds to the ideal projection Lagrange interpolation operator called Lagrange projection operator.2005, de Boor mentioned the following questions in his review: the ideal interpolation error expression is the ideal structure of projection operator is unified; second: what is the "good" ideal projection operator error formula; third: if an ideal projection operator for Hermite projection operator, how to calculate the approximation of Lagrange projection operator of it. So far, the problem is still a hot topic in the ideal interpolation. For simplicity, we call the first two questions for the problem of the ideal interpolation error formula, called the last question For the discretization of the ideal interpolation. The knowledge of algebraic geometry theory this paper will use, some theoretical results are also given. The main work is as follows: 1. gives the error structure expression of ideal projection operator. A unified structure error ideal projection operator is simple is beautiful. Which could be generalized to the multivariate case, de Boor put forward the concept of the ideal projection operator of "good" error formula. "Good" is a kind of error formula of error structure expression, specifically, refers to the existence of homogeneous polynomial Hj and linear operator Q so that the interpolation error can be expressed as f-Pf = j=1mCj (Hj (D) F) HJ and orthogonal Hj (D) hk= J, K F, which is inserted polynomial function, P is the ideal projection operator, MA (D) for the differential operator, {h1,..., hm} is the ideal Boor ideal kerP based de had to guess all the ideal projection operator has "good" error formula But then, Shekhtman gave a counterexample of two yuan case, and asserts that the most ideal projection operator is not "good" error formula. We study the algebraic structure of the error formula of ideal projection operator, based on "good" error formula, puts forward the concept of "a" type of error formula and then use the ideal reduction theory, proved that the reduced Grobner based kernel space all ideal projection operator in lexicographic order under the support "general" type error formula. Finally using B spline theory, gives the Shekhtman a counterexample "general" type error formula of the specific expression of.2. presents a Lagrange projection operator "good" error formula of specific expression. We mentioned is not all ideal projection operator has good error formula. Until now, the people of the "good" error formula of the existence of research The progress of Shekhtman proved that the ideal projection operator special geometric distribution node has a "good" error formula, "good" error formula of ideal projection operator Li U after our Qian Le robner condyle of Worcestershire based, de Boor provides a specific expression with tensor product nodes and meet the Lagrange GC node projection operator the "good" error formula. Inspired by their work, we study the error formula of projection operator of a special kind of ideal for Lagrange projection operator Cartesian points on the first use of difference algorithm, is presented. Then the interpolation remainder interpolation remainder sorting into difference quotient "good" error formula. Finally the relationship the difference with the spline integral, the discrete problem specific expression.3. "error formula is studied for a class of two yuan Hermite projection operator is given. When people promote a concept, The general structure will retain the properties of the original. In one yuan case, Hermite interpolation is an extreme form of Lagrange interpolation. This fact inspired de Boor definition of Hermite projection operator to limit Lagrange projection operator. Although the ideal projection operator is Hermite projection operator, and this conclusion in some multivariate case was also established, but there are examples of multiple ideal projection operator in non Hermite. So the judge an ideal projection operator is Hermite projection operator, and how to calculate the Lagrange projection approximation of Hermite projection operator column is the problem that people cares very. Around this problem (discrete ideal interpolation), De, Boor and Shekhtman proved that two the ideal element projection operator is Hermite projection operator, a method of calculating the corresponding Lagrange projection operator sequence is given. But the method itself high complexity, Not easy to implement. We study a special class of two yuan Hermite projection operator, the interpolation condition for the delta zeta functional / P (n) (D), P (n): =Fn[x, y] (?) spanF{pn}, which is at the point of the delta zeta zeta functional assignment, D differential operator, Fn[x. Two yuan is smaller than the number of polynomial y] n collection, PN to two yuan n polynomial. For such an arbitrary Hermite projection operator, we give a method of calculating the Lagrange projection operator, the method is simple and effective and almost without any cost.

【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.3

【相似文献】