多元样条及其某些应用
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摘要
计算几何是一门新兴的几何分支学科,是几何学、计算数学与计算机科学的交叉学科.样条是计算几何的基本理论工具和基础.自1946年I. J. Schoenberg建立了一元样条的基础以来,多元样条的研究一直没有取得实质性的进展.1975年,王仁宏教授开创了以光滑余因子协调方法为核心的多元样条的代数几何方法,得到了任意剖分下多元样条的充分必要条件,把任意剖分下的多元样条归结为求解以多项式为系数的方程组,从而建立了任意剖分下多元样条的基本理论框架.本文主要利用研究多元样条的光滑余因子协调方法,对多元样条的理论及应用进行了研究.本文的主要工作如下：
1.本文回顾了王仁宏教授建立的研究多元样条的基本理论框架,介绍了光滑余因子协调方法及其在相关领域中取得的成果.
2.1980年,G.Farin在多元样条的基本理论框架下考虑了三角形剖分上的的分片多项式的：Bezier坐标和光滑拼接条件之间的关系,建立了B网方法的理论基础.需要指出的是,B网方法只适用于研究单纯形剖分上的多元样条,而光滑余因子协调方法可以进行任意剖分上的多元样条的研究,本文证明了这样一个事实：B网方法是建立在王仁宏教授建立的研究多元样条的基本理论框架之上的,它可由光滑余因子协调方法直接导出.
3.利用多元样条进行散乱数据拟合是计算几何中一个非常重要的课题.本文讨论了2-型三角剖分上的2元样条空间S21(△m,n(2))的结构和性质,并利用S21(△m,n(2))中的基函数对给定的散乱数据点进行拟合,提出了BS2算法.该算法避开了适定结点组的选取.数据实验表明BS2算法能够达到较好的拟合效果.为了构造出精确度更高,光顺性更好的拟合函数,本文对BS2算法进行了改进：提出了一种新的算法：MBS算法.该算法利用多重拟合的思想,对前一步用BS2算法得到的拟合函数的误差进行拟合,这样形成了一组拟合函数,最终的拟合函数是这组拟合函数的和.
4.本文利用2-型剖分上带有边界条件的样条空间S21;0(△m,n(2))和S42,3;0(△m,n(2))对线性抛物型方程求数值解,并提出了一种自适应的非连续伽略金有限元方法(简称ADB方法).数据实验表明,利用ADB方法得到的线性抛物型方程的数值解具有较高的精确度.样条函数基底具有对称性,可存储性和可计算性等优点,这些优点使得ADB方法在实现上也更为方便.
Computational geometry is a new subject of geometry. It is a interdisciplinary subject of geometry, computational mathematics and computer science. The spline is a basic theoretical tool of computational geometry. After I. J. Schoenberg established the theory of the univariate spline in 1946, the research on the multivariate spline does not improve essentially. In 1975, Prof. Renhong Wang established the algebraic geometry method for the multivariate spline using the so-called conforamlity method of smoothing cofactor (CSC method). Then the sufficient and necessary conditions for multivariate splines over arbitrary partitions are obtained. The problems of multivariate splines over arbitrary partitions turn to solving a system of equations with polynomials as coefficients, and the theoretical framework of multivariate splines is established. This article uses the conformality method of smoothing cofactor to discuss the theories and applications of multivariate splines. The main work is as follows:
1. We retrospect the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, and discuss the conformality method of smoothing cofactor and his advanced products about the research on the space of multivariate splines.
2. In 1980, under the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, G. Farin considered the relationship between the barycentric coordinates of a piecewise polynomial over two adjacent triangles and smoothness con-ditions. He introduced the B-net method which is suitable for studying the multivariate splines over simplex partitions. We know that, the CSC method is an approach to study multivariate splines over arbitrary partition. So there is some relationship between the CSC method and the B-net method. This article indicates that the B-net method is established the basic theoretical framework of multivariate splines established by Prof. Renhong Wang. It could be derived directly by the CSC method.
3. Scattered data approximation is a very important issue in computational geome-try. This article discusses the structure and nature of spline space (?). We proposed an algorithm (named BS2 Algorithm), which uses the bases in (?)to fitting the given scattered data points. The BS2 Algorithm avoids the selection of the Lagrange interpo-lation set. The numerical results indicate that the BS2 Algorithm has fine accuracy. In order to obtain a approximation function with higher accuracy and finer smoothness, we modify the BS2 Algorithm and propose a new algorithm:MBS Algorithm. It makes use of a hierarchy of 2-type partitions to generate a sequence of functions whose sum approaches the desired approximation functions.
4. Using 2-type partitions spline space with boundary conditions, (?) and (?), this article proposes a numerical method to solve linear parabolic equations. We give an adaptive discontinuous Galerkin finite element method (the ADB method). The numerical results show that the solutions obtained by the ADB method have high accuracy. Since the bases of (?) and (?) are easy to store and evaluate, the ADB method is more convenient to implement.
1.本文回顾了王仁宏教授建立的研究多元样条的基本理论框架,介绍了光滑余因子协调方法及其在相关领域中取得的成果.
2.1980年,G.Farin在多元样条的基本理论框架下考虑了三角形剖分上的的分片多项式的：Bezier坐标和光滑拼接条件之间的关系,建立了B网方法的理论基础.需要指出的是,B网方法只适用于研究单纯形剖分上的多元样条,而光滑余因子协调方法可以进行任意剖分上的多元样条的研究,本文证明了这样一个事实：B网方法是建立在王仁宏教授建立的研究多元样条的基本理论框架之上的,它可由光滑余因子协调方法直接导出.
3.利用多元样条进行散乱数据拟合是计算几何中一个非常重要的课题.本文讨论了2-型三角剖分上的2元样条空间S21(△m,n(2))的结构和性质,并利用S21(△m,n(2))中的基函数对给定的散乱数据点进行拟合,提出了BS2算法.该算法避开了适定结点组的选取.数据实验表明BS2算法能够达到较好的拟合效果.为了构造出精确度更高,光顺性更好的拟合函数,本文对BS2算法进行了改进：提出了一种新的算法：MBS算法.该算法利用多重拟合的思想,对前一步用BS2算法得到的拟合函数的误差进行拟合,这样形成了一组拟合函数,最终的拟合函数是这组拟合函数的和.
4.本文利用2-型剖分上带有边界条件的样条空间S21;0(△m,n(2))和S42,3;0(△m,n(2))对线性抛物型方程求数值解,并提出了一种自适应的非连续伽略金有限元方法(简称ADB方法).数据实验表明,利用ADB方法得到的线性抛物型方程的数值解具有较高的精确度.样条函数基底具有对称性,可存储性和可计算性等优点,这些优点使得ADB方法在实现上也更为方便.
Computational geometry is a new subject of geometry. It is a interdisciplinary subject of geometry, computational mathematics and computer science. The spline is a basic theoretical tool of computational geometry. After I. J. Schoenberg established the theory of the univariate spline in 1946, the research on the multivariate spline does not improve essentially. In 1975, Prof. Renhong Wang established the algebraic geometry method for the multivariate spline using the so-called conforamlity method of smoothing cofactor (CSC method). Then the sufficient and necessary conditions for multivariate splines over arbitrary partitions are obtained. The problems of multivariate splines over arbitrary partitions turn to solving a system of equations with polynomials as coefficients, and the theoretical framework of multivariate splines is established. This article uses the conformality method of smoothing cofactor to discuss the theories and applications of multivariate splines. The main work is as follows:
1. We retrospect the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, and discuss the conformality method of smoothing cofactor and his advanced products about the research on the space of multivariate splines.
2. In 1980, under the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, G. Farin considered the relationship between the barycentric coordinates of a piecewise polynomial over two adjacent triangles and smoothness con-ditions. He introduced the B-net method which is suitable for studying the multivariate splines over simplex partitions. We know that, the CSC method is an approach to study multivariate splines over arbitrary partition. So there is some relationship between the CSC method and the B-net method. This article indicates that the B-net method is established the basic theoretical framework of multivariate splines established by Prof. Renhong Wang. It could be derived directly by the CSC method.
3. Scattered data approximation is a very important issue in computational geome-try. This article discusses the structure and nature of spline space (?). We proposed an algorithm (named BS2 Algorithm), which uses the bases in (?)to fitting the given scattered data points. The BS2 Algorithm avoids the selection of the Lagrange interpo-lation set. The numerical results indicate that the BS2 Algorithm has fine accuracy. In order to obtain a approximation function with higher accuracy and finer smoothness, we modify the BS2 Algorithm and propose a new algorithm:MBS Algorithm. It makes use of a hierarchy of 2-type partitions to generate a sequence of functions whose sum approaches the desired approximation functions.
4. Using 2-type partitions spline space with boundary conditions, (?) and (?), this article proposes a numerical method to solve linear parabolic equations. We give an adaptive discontinuous Galerkin finite element method (the ADB method). The numerical results show that the solutions obtained by the ADB method have high accuracy. Since the bases of (?) and (?) are easy to store and evaluate, the ADB method is more convenient to implement.
引文
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[2]王仁宏.多元齿的结构与插值.数学学报,1975,18：91-106.
[3]王仁宏.施锡泉,罗钟铉,苏志勋.多元样条函数及其应用.北京：科学出版社,1994.
[4]Wang R H. Multivariate Spline Functions and Their Applications. Beijing/New York: Science Press/Kluwer Pub.,1994/2001.
[5]王仁宏.任意剖分下的多元样条分析.中国科学,数学专辑I,1979：215-226.
[6]王仁宏.任意剖分下的多元样条分析ii-空间形式.高等学校计算数学学报,1980,1：78-81.
[7]Wang R H. The dimension and basis of spaces of multivariate spline. Journal of Computational and Applied Mathematics,1985,12-13:163-177.
[8]Wang R H. Multivariate spline and algebraic geometry. Journal of Computational and Applied Mathematics,2000,121:153-163.
[9]王仁宏.多元弱样条.广州计算数学会议,1987.
[10]王仁宏,何天晓.不均匀第二型三角剖分下的带有边界条件的样条函数空间.科学通报：1985,4：249-251.
[11]王仁宏,卢旭光.关于三角剖分下二元样条空间的维数.中国科学(A辑),1988,1：585-594.
[12]王仁宏.关于多元样条空间的维数.科学通报,1988,6：473-474.
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