基于二元Box样条的一种逼近格式的研究

发布时间：2018-05-22 17:20

本文选题：Box样条 + 插值　；参考：《吉林大学》2017年硕士论文

【摘要】：逼近论是计算数学领域的一个重要分支,在理论研究和实际应用中都有着重要意义,同时逼近论与代数、微分方程等其他数学学科也有着非常密切的联系.本文讨论逼近论中的二元情形,我们知道二元Box样条函数是一元B样条函数的多元推广,同时二元Box样条是多元逼近问题中的一类非常重要的基函数.本文将文献[1]中以一元B样条为基函数的逼近格式推广到二元情形,即以二元Box样条函数为基函数构造二元函数的逼近算子,同时可以通过对得到的逼近算子求导的方式来逼近二元函数的空间偏导数.为了得到具有一定代数精度的插值算子,本文分两个阶段来构造二元逼近算子:第一阶段是以二元Box样条为基函数,分别构造一个具有紧支集的插值算子L(不具有多项式再生性)与一个具有一定代数精度的拟插值算子Q;第二阶段是将第一阶段构造的插值算子L和拟插值算子Q做布尔和,生成一个新的逼近算子l,该算子既有与拟插值算子Q相同次数的代数精度又具有插值性质.通过对二元逼近算子l求导还可以逼近二元函数的空间偏导数,本文分别以二元乘积型Box样条和(2,2,2)阶Box样条N(2,2,2)(x)为基函数来构造上述的逼近算子l,并在本文的第四章通过数值实验进一步说明逼近算子l对二元函数的逼近性质.本文共分五章:第一章是本文的绪论部分,第一节主要介绍了逼近论的起源、发展,第二节介绍本文的主要研究思路.第二章第一节从两个角度给出二元Box样条函数的定义,第二节介绍了二元Box样条的几个重要性质,包括二元Box样条的次数、光滑度、支集性质、非负单位分解性以及中心对称性等.第三章是本文的核心章节,第一节介绍[1]中以一元B样条为基函数的逼近格式,第二节介绍本文的以二元Box样条为基函数的二元逼近算子的一般格式.后面两节则分别给出以二元乘积型Box样条和(2,2,2)阶Box样条N(2,2,2)(x)为基函数的逼近算子lm的具体格式.第四章数值实验,第一节验证了以(2,2,2)阶Box样条为基函数的逼近算子l(2,2,2)的代数精度.具体做法,分别以二元单项式f1=xy,f2 = x2,f2 =xy2作为被逼近函数,通过逼近算子l(2,2,2)的图像与原函数图像的对比,更直观地说明了 l2,2,2)对二元单项式f1=xy,f2=x2y,f3= =xy2是精确逼近的,即l(2,2,2)的代数精度为3.对于不能被l(2,2,2)精确逼近的二元函数,我们对逼近算子l(2,2,2)做伸缩变换,即改变插值节点的步长h,通过观察逼近误差随h的变化进一步说明 l(2,2,2)的逼近能力.第五章结论,对本文的主要思想进行总结,并提出本文的不足.
[Abstract]:Approximation theory is an important branch in the field of computational mathematics, which is of great significance in both theoretical research and practical application. At the same time, approximation theory is closely related to algebra, differential equations and other mathematical disciplines. In this paper, we discuss the binary case in approximation theory. We know that the binary Box spline function is a multivariate generalization of the one-variable B-spline function, and that the binary Box spline is a very important basis function in the problem of multivariate approximation. In this paper, we extend the approximation scheme of one-variable B-spline function in reference [1] to the binary case, that is, using the binary Box spline function as the basis function to construct the approximation operator of the binary function. At the same time, the spatial partial derivatives of binary functions can be approximated by the derivation of the obtained approximation operators. In order to obtain interpolation operators with certain algebraic precision, this paper constructs binary approximation operators in two stages: the first stage is based on binary Box splines. We construct an interpolation operator L (without polynomial reproducing) with compact support set and a quasi interpolation operator Q with a certain algebraic precision, the second stage is to do Boolean sum of the interpolation operator L and the quasi interpolation operator Q, which are constructed in the first stage. A new approximation operator l is generated, which has the algebraic accuracy and interpolation property of the same degree as the quasi interpolation operator Q. By derivation of bivariate approximation operator l, the spatial partial derivative of bivariate function can be approximated. In this paper, the Box splines of binary product type and the Box splines of order Box are used as basis functions to construct the above approximation operators. In the fourth chapter of this paper, the approximation properties of approximation operators l to binary functions are further explained by numerical experiments. This paper is divided into five chapters: the first chapter is the introduction of this paper, the first section mainly introduces the origin and development of approximation theory, the second section introduces the main research ideas of this paper. In the second chapter, we give the definition of binary Box spline function from two angles. In the second section, we introduce some important properties of binary Box spline, including the degree, smoothness and support property of binary Box spline. Nonnegative unit decomposition and central symmetry. The third chapter is the core chapter of this paper. In section 1, we introduce the approximation scheme of B-spline as basis function in [1], and the general format of binary approximation operator with binary Box spline as basis function in the second section. In the latter two sections, the approximation operator lm of the Box splines of order N ~ (2 ~ (2) ~ (2) ~ (2) ~ (2) is given, respectively, in which the Box splines of the bivariate product type and the Box splines of order N _ (2 ~ (2) ~ (2) are taken as the basis functions. In the fourth chapter, numerical experiments are conducted. In the first section, we verify the algebraic accuracy of the approximation operator ln ~ 2 ~ 2 ~ 2 ~ 2 ~ (2), which takes the Box spline of order 2 as the basis function. In this paper, we take the binary monomial form f _ 1C _ XY _ 2 = x _ 2o _ f _ 2 as the approximated function, and by comparing the image of the approximation operator l ~ (2 +) ~ (2) ~ (2) with the original image, it is more intuitively shown that the binary monomial form f _ 1 / f _ 2x _ 2yf _ 3 = xy2 is more intuitively approximate to the binary monomial form f _ 1 / f _ 2x _ 2y _ 2yf _ 3 = xy2. The algebraic accuracy is 3. For binary functions which can not be accurately approximated by L2 / 2), we make a telescopic transformation on the approximation operator lt2 / 2 / 2), that is, to change the step size of the interpolated node, and to further illustrate the approximation ability of ln2m2m2k2) by observing the variation of the approximation error with h. The fifth chapter summarizes the main ideas of this paper, and puts forward the shortcomings of this paper.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.5

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