解稀疏插值问题的代数几何方法

发布时间：2018-06-15 22:52

本文选题：多项式方程组 + 同伦方法　；参考：《大连理工大学》2016年博士论文

【摘要】：插值是计算数学中的一个基本问题,在科学与工程很多领域有重要应用.其中,稀疏插值问题是一类有趣的、有重要应用背景但相对来说研究还不够成熟的问题,近年来受到越来越多的国内外学者的关注.多项式方程组求解问题自古以来就是一个重要并且困难的问题,是代数学、代数几何、计算数学与计算机数学的重要研究课题.本文研究由稀疏插值问题及与其密切相关的具有高度振荡系数的线性椭圆型微分方程数值解中衍生出来的多项式方程组的解的性质和高效率的解法.第一章简要地介绍了稀疏插值问题的发展和应用以及解多项式方程组的同伦方法的一些进展.第二章研究等距稀疏插值问题.对一般的采样数据,我们证明了具有2n个等距采样点的稀疏插值问题所衍生出来的多项式方程组恰好具有n!个非奇异孤立解,并且它们都属于同一个等价类.利用该性质,我们给出了一种高效率的系数参数同伦方法.该算法在第一阶段不需要任何计算量,第二阶段仅需要跟踪一条路径即可求得该多项式方程组的全部孤立解.在第三章,对一般的多项式方程组,在给定变元分组下,我们证明了当多项式方程组的最高次齐次部分只有平凡解时,其孤立解的个数等于该变元分组所对应的多重齐次Bezout数.本章是第四章关于带跳点的等距稀疏插值问题所衍生出来的多项式方程组孤立解的性质研究的理论基础.第四章研究带跳点的等距稀疏插值问题.对带跳点的等距稀疏插值问题所衍生出来的多项式方程组,我们给出了一个关于其孤立解个数和解的等价类个数的猜想,并对部分情形,通过消元化简后用同伦方法证明了该猜想.随后,我们给出求该多项式方程组全部孤立解的高效的系数参数同伦方法.该算法在第一阶段只需很小的计算量,第二阶段所需跟踪的同伦路径的条数与解的等价类的个数相等,远远小于孤立解的个数.第五章研究具有高度振荡系数的线性椭圆型微分方程的稀疏解.与传统数值算法(如谱方法、有限元等)不同,基于真解可用很少几个具有较大权值系数的基函数的线性组合来很好地逼近的观察,我们采用不定基函数的离散化策略.这样,与稀疏插值问题类似,该问题可以归结为一类小规模的具有特殊结构的多项式方程组求解问题,而不是一个较大规模的线性问题.在此基础上,我们给出了求解该问题的高效的数值算法.此外,振荡性的增强不会改变该数值算法中所需要的基函数的数量.
[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.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O174.42

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