發(fā)布時(shí)間：2018-05-22 17:20

  本文選題：Box樣條 + 插值　； 參考：《吉林大學(xué)》2017年碩士論文

【摘要】：逼近論是計(jì)算數(shù)學(xué)領(lǐng)域的一個(gè)重要分支,在理論研究和實(shí)際應(yīng)用中都有著重要意義,同時(shí)逼近論與代數(shù)、微分方程等其他數(shù)學(xué)學(xué)科也有著非常密切的聯(lián)系.本文討論逼近論中的二元情形,我們知道二元Box樣條函數(shù)是一元B樣條函數(shù)的多元推廣,同時(shí)二元Box樣條是多元逼近問題中的一類非常重要的基函數(shù).本文將文獻(xiàn)[1]中以一元B樣條為基函數(shù)的逼近格式推廣到二元情形,即以二元Box樣條函數(shù)為基函數(shù)構(gòu)造二元函數(shù)的逼近算子,同時(shí)可以通過對得到的逼近算子求導(dǎo)的方式來逼近二元函數(shù)的空間偏導(dǎo)數(shù).為了得到具有一定代數(shù)精度的插值算子,本文分兩個(gè)階段來構(gòu)造二元逼近算子:第一階段是以二元Box樣條為基函數(shù),分別構(gòu)造一個(gè)具有緊支集的插值算子L(不具有多項(xiàng)式再生性)與一個(gè)具有一定代數(shù)精度的擬插值算子Q;第二階段是將第一階段構(gòu)造的插值算子L和擬插值算子Q做布爾和,生成一個(gè)新的逼近算子l,該算子既有與擬插值算子Q相同次數(shù)的代數(shù)精度又具有插值性質(zhì).通過對二元逼近算子l求導(dǎo)還可以逼近二元函數(shù)的空間偏導(dǎo)數(shù),本文分別以二元乘積型Box樣條和(2,2,2)階Box樣條N(2,2,2)(x)為基函數(shù)來構(gòu)造上述的逼近算子l,并在本文的第四章通過數(shù)值實(shí)驗(yàn)進(jìn)一步說明逼近算子l對二元函數(shù)的逼近性質(zhì).本文共分五章:第一章是本文的緒論部分,第一節(jié)主要介紹了逼近論的起源、發(fā)展,第二節(jié)介紹本文的主要研究思路.第二章第一節(jié)從兩個(gè)角度給出二元Box樣條函數(shù)的定義,第二節(jié)介紹了二元Box樣條的幾個(gè)重要性質(zhì),包括二元Box樣條的次數(shù)、光滑度、支集性質(zhì)、非負(fù)單位分解性以及中心對稱性等.第三章是本文的核心章節(jié),第一節(jié)介紹[1]中以一元B樣條為基函數(shù)的逼近格式,第二節(jié)介紹本文的以二元Box樣條為基函數(shù)的二元逼近算子的一般格式.后面兩節(jié)則分別給出以二元乘積型Box樣條和(2,2,2)階Box樣條N(2,2,2)(x)為基函數(shù)的逼近算子lm的具體格式.第四章數(shù)值實(shí)驗(yàn),第一節(jié)驗(yàn)證了以(2,2,2)階Box樣條為基函數(shù)的逼近算子l(2,2,2)的代數(shù)精度.具體做法,分別以二元單項(xiàng)式f1=xy,f2 = x2,f2 =xy2作為被逼近函數(shù),通過逼近算子l(2,2,2)的圖像與原函數(shù)圖像的對比,更直觀地說明了 l2,2,2)對二元單項(xiàng)式f1=xy,f2=x2y,f3= =xy2是精確逼近的,即l(2,2,2)的代數(shù)精度為3.對于不能被l(2,2,2)精確逼近的二元函數(shù),我們對逼近算子l(2,2,2)做伸縮變換,即改變插值節(jié)點(diǎn)的步長h,通過觀察逼近誤差隨h的變化進(jìn)一步說明 l(2,2,2)的逼近能力.第五章結(jié)論,對本文的主要思想進(jìn)行總結(jié),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.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前3條

1 徐利治,楊家新;多元函數(shù)逼近研究近況述評[J];數(shù)學(xué)進(jìn)展;1987年03期

2 王仁宏;梁學(xué)章;;二元連續(xù)函數(shù)在某些多項(xiàng)式類上的一致逼近理論[J];吉林大學(xué)自然科學(xué)學(xué)報(bào);1965年02期

3 王仁宏;;高維歐氏空間上線性正算子對多元函數(shù)的逼近枎之漸近公式[J];吉林大學(xué)自然科學(xué)學(xué)報(bào);1964年01期

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本文編號(hào)：1922955