本文選題：多項(xiàng)式方程組 + 同倫方法�。� 參考：《大連理工大學(xué)》2016年博士論文

【摘要】：插值是計算數(shù)學(xué)中的一個基本問題,在科學(xué)與工程很多領(lǐng)域有重要應(yīng)用.其中,稀疏插值問題是一類有趣的、有重要應(yīng)用背景但相對來說研究還不夠成熟的問題,近年來受到越來越多的國內(nèi)外學(xué)者的關(guān)注.多項(xiàng)式方程組求解問題自古以來就是一個重要并且困難的問題,是代數(shù)學(xué)、代數(shù)幾何、計算數(shù)學(xué)與計算機(jī)數(shù)學(xué)的重要研究課題.本文研究由稀疏插值問題及與其密切相關(guān)的具有高度振蕩系數(shù)的線性橢圓型微分方程數(shù)值解中衍生出來的多項(xiàng)式方程組的解的性質(zhì)和高效率的解法.第一章簡要地介紹了稀疏插值問題的發(fā)展和應(yīng)用以及解多項(xiàng)式方程組的同倫方法的一些進(jìn)展.第二章研究等距稀疏插值問題.對一般的采樣數(shù)據(jù),我們證明了具有2n個等距采樣點(diǎn)的稀疏插值問題所衍生出來的多項(xiàng)式方程組恰好具有n!個非奇異孤立解,并且它們都屬于同一個等價類.利用該性質(zhì),我們給出了一種高效率的系數(shù)參數(shù)同倫方法.該算法在第一階段不需要任何計算量,第二階段僅需要跟蹤一條路徑即可求得該多項(xiàng)式方程組的全部孤立解.在第三章,對一般的多項(xiàng)式方程組,在給定變元分組下,我們證明了當(dāng)多項(xiàng)式方程組的最高次齊次部分只有平凡解時,其孤立解的個數(shù)等于該變元分組所對應(yīng)的多重齊次Bezout數(shù).本章是第四章關(guān)于帶跳點(diǎn)的等距稀疏插值問題所衍生出來的多項(xiàng)式方程組孤立解的性質(zhì)研究的理論基礎(chǔ).第四章研究帶跳點(diǎn)的等距稀疏插值問題.對帶跳點(diǎn)的等距稀疏插值問題所衍生出來的多項(xiàng)式方程組,我們給出了一個關(guān)于其孤立解個數(shù)和解的等價類個數(shù)的猜想,并對部分情形,通過消元化簡后用同倫方法證明了該猜想.隨后,我們給出求該多項(xiàng)式方程組全部孤立解的高效的系數(shù)參數(shù)同倫方法.該算法在第一階段只需很小的計算量,第二階段所需跟蹤的同倫路徑的條數(shù)與解的等價類的個數(shù)相等,遠(yuǎn)遠(yuǎn)小于孤立解的個數(shù).第五章研究具有高度振蕩系數(shù)的線性橢圓型微分方程的稀疏解.與傳統(tǒng)數(shù)值算法(如譜方法、有限元等)不同,基于真解可用很少幾個具有較大權(quán)值系數(shù)的基函數(shù)的線性組合來很好地逼近的觀察,我們采用不定基函數(shù)的離散化策略.這樣,與稀疏插值問題類似,該問題可以歸結(jié)為一類小規(guī)模的具有特殊結(jié)構(gòu)的多項(xiàng)式方程組求解問題,而不是一個較大規(guī)模的線性問題.在此基礎(chǔ)上,我們給出了求解該問題的高效的數(shù)值算法.此外,振蕩性的增強(qiáng)不會改變該數(shù)值算法中所需要的基函數(shù)的數(shù)量.
[Abstract]:Interpolation is a basic problem in computational mathematics and has important applications in many fields of science and engineering. Among them, sparse interpolation is a kind of interesting problem, which has important application background, but the research is not mature enough, and has been paid more and more attention by domestic and foreign scholars in recent years. The problem of solving polynomial equations has been an important and difficult problem since ancient times. It is an important research topic in algebra, algebraic geometry, computational mathematics and computer mathematics. In this paper, we study the properties and efficient solutions of polynomial equations derived from sparse interpolation problems and the numerical solutions of linear elliptic differential equations with highly oscillatory coefficients. In chapter 1, the development and application of sparse interpolation problem and the homotopy method for solving polynomial equations are briefly introduced. In chapter 2, we study the equidistant sparse interpolation problem. For the general sampling data, we prove that the polynomial equations derived from the sparse interpolation problem with 2n equidistant sampling points have exactly n! And they all belong to the same equivalence class. By using this property, we give an efficient homotopy method of coefficient parameters. The algorithm does not require any computation in the first stage. In the second stage, only one path is followed to obtain all the isolated solutions of the polynomial equations. In chapter 3, for general polynomial equations, we prove that when the highest homogeneous part of polynomial equations has only trivial solution, the number of isolated solutions is equal to the multiple homogeneous Bezout number corresponding to the variable group. This chapter is the theoretical basis of the fourth chapter on the properties of solitary solutions of polynomial equations derived from the equidistant sparse interpolation problem with jump points. In chapter 4, we study the equidistant sparse interpolation with hopping points. For the polynomial equations derived from the equidistant sparse interpolation problem with hopping points, we give a conjecture about the number of isolated solutions and the number of equivalent classes, and for some cases, The conjecture is proved by homotopy method after simplifying by elimination. Then we give an efficient homotopy method for finding all isolated solutions of the polynomial equations. The number of homotopy paths tracked in the second stage is equal to the number of equivalent classes of solutions, which is far less than the number of isolated solutions. In chapter 5, the sparse solutions of linear elliptic differential equations with high oscillation coefficients are studied. Different from traditional numerical algorithms (such as spectral method, finite element method, etc.), based on the observation that only a few linear combinations of basis functions with large weight coefficients can be used for proper solutions, we adopt the discretization strategy of indefinite basis functions. In this way, similar to the sparse interpolation problem, the problem can be reduced to a small problem of solving polynomial equations with special structure, rather than a large scale linear problem. On this basis, we give an efficient numerical algorithm to solve the problem. Furthermore, the enhancement of oscillation does not change the number of basis functions required in the numerical algorithm.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O174.42

【相似文獻(xiàn)】

相關(guān)期刊論文 前10條

1 張作順;多元基插值問題[J];數(shù)學(xué)年刊A輯(中文版);1989年05期

2 謝四清;在π_n(x)零點(diǎn)上的(0,1,3)插值[J];數(shù)學(xué)研究與評論;2001年02期

3 吳化璋,盛金苗;一類插值問題的統(tǒng)一研究方法(英文)[J];黑龍江大學(xué)自然科學(xué)學(xué)報;2004年04期

4 顏寧生;一類對稱插值[J];北京服裝學(xué)院學(xué)報;2005年03期

5 吳化璋,喬云;Nevanlinna-Pick插值問題與相關(guān)冪矩量問題之間的聯(lián)系(英文)[J];中國科學(xué)技術(shù)大學(xué)學(xué)報;2005年01期

6 張慧;王德義;;推廣的(0,p(D))整插值[J];榆林學(xué)院學(xué)報;2006年06期

7 黃鴻慈;關(guān)于插值的穩(wěn)定性[J];計算數(shù)學(xué);1982年02期

8 張有訓(xùn);王輝;;二元二次分片多項(xiàng)式的第Ⅱ類插值問題[J];浙江師范學(xué)院學(xué)報(自然科學(xué)版);1985年02期

9 鄔弘毅;;關(guān)于四次缺插值祥條的適定性問題[J];安徽工學(xué)院學(xué)報;1987年Z1期

10 何偉保;用非多項(xiàng)式樣條解兩類插值問題[J];貴州工學(xué)院學(xué)報;1994年02期

相關(guān)會議論文 前3條

1 雷娜;滕苑;任玉雪;;Hermite插值的幾何求基降秩算法[A];第六屆全國幾何設(shè)計與計算學(xué)術(shù)會議論文集[C];2013年

2 張衛(wèi)祥;;自適應(yīng)抽樣與基于MQ-B-樣條的徑向函數(shù)插值[A];幾何設(shè)計與計算的新進(jìn)展[C];2005年

3 雷娜;崔凱;;多元Birkhoff插值問題的穩(wěn)定單項(xiàng)基[A];第六屆全國幾何設(shè)計與計算學(xué)術(shù)會議論文集[C];2013年

相關(guān)博士學(xué)位論文 前6條

1 王倩;球面曲線插值問題及不變量的研究與應(yīng)用[D];大連理工大學(xué);2016年

2 焦力賓;解稀疏插值問題的代數(shù)幾何方法[D];大連理工大學(xué);2016年

3 王筱穎;多元多項(xiàng)式插值問題的牛頓基[D];吉林大學(xué);2010年

4 陳濤;多元多項(xiàng)式插值的極小次數(shù)牛頓基[D];吉林大學(xué);2007年

5 陳志祥;基于Chebyshev多項(xiàng)式零點(diǎn)的若干實(shí)插值問題[D];浙江大學(xué);2002年

6 李U，

本文編號：2023968