Bivariate Spline Method For Scattered Data Fitting

Spline is an excellent tool for numerical approximation,which is a perfect result of the combination of approximation theory and computer theory.In modern computational work,spline is the major tool for engineers and CAD software since it is easy to represent and efficient to evaluate.The spline finite element is also an efficient tool in finite element analysis.The theory of univariate splines began its rapid development in the early sixties, resulting in several thousand research papers and a number of books.This development was largely over by 1980,and the bulk of what is known today was treated already in the classic monographs of deBoor[10]and Sehumaker[31].Univariate splines have become an essential tool in a wide variety of application areas, and are by now a standard topic in numerical analysis books.If 1960-1980 was the age of univariate splines,then now can be regarded as the age of multivariate splines. Prior to 1980 there were some results for using piccewisc polynomials in two and three variables in the finite element method,but multivariate splines had attracted relatively little attention.Now we have thousands of papers on the subject.The main purpose of this thesis is to discuss about multivariate spline method(especially in bivariate spline) and its application in scattered data fitting.Here a multivariate spline is a function which is made up of pieces of polynomials defined on some partition△of setΩ,and joined together to ensure some degree of global smoothness.As we shall see,multivariate polynomial splines have many of the same features which make the univariate splincs such powerful tools for applications such as splines are easy to work with computationally,and there are stable and efficient algorithms for evaluating their derivatives and integrals,ect.This thesis is organized as follows:In the first chapter we introduce the spline in B-form and some of its useful algorithm relate to triangle such as the de casteljau algorithm,derivative algorithm, integrals algorithm,etc.Some of them are very important in building the matrix. Then we show how to use the smooth condition and build the whole space of spline. Next we list some different kinds of triangulations which are usually used in many papers.Finally,we introduce a iteration algorithm used in matrix computation.In the second chapter we discuss about scattered data hermite interpolation problem.We start this from how to build a spline which satisfied the interpolation problem.We provide a new energy function method which is different from[26]and [29].Then we show the existence and uniqueness of the spline which is generated by our method.Next,we present the error bound of the spline.Finally,we offer some numerical experiments to demonstrate our method.The last two chapters are all about scattered data fitting.If the number of the data are large or very small,it may not be appropriate to interpolate the data.In the third chapter we use extended penalty function method to fit the data when the number of the data are very small.This method can deal with the data with derivative compare to penalty function method.In the final chapter,we use weighted least squares method to fit the data of which the number is large.We can adjust the weighted functions according to the importance of each data.As always,we show the existence and uniqueness of the splines in last two chapters and also with error bound and numerical experiments as well.