散乱数据拟合的模型、方法和理论(第二版)(英文版)

Chapter 1 Scattered Data Approximation and Multivariate Polynomial Interpolation 1.1　Motivation Problems Functions studied in mathematics have been widely used to model various practical problems. For example， one can utilize a linear function (a straight line) to describe the trajectory of light. The physics law says that light travels along a straight line in vacuum without the impact of a magnetic field. However， light is not often in vacuum. Besides light in fact is impacted by the magnetic field of the earth and the sun. Therefore， the description that light travels along a straight line is only an approximation. Such an approximation is well accepted because the trajectory of light in our daily life is so close to a straight line that the difference between the true trajectory and the straight line is negligible. Strictly speaking， two reasons make such an approximation acceptable. Firstly， the accurate description of the trajectory of light is almost impossible. Secondly， the straight line approximation to the trajectory of light， though simple， is sufficiently accurate. It is unnecessary to use nonlinear and complex functions to describe the trajectory of light. Such an example suggests two principles for general approximation problems. Firstly， it is better to choose a simple and useful function to approximate our target problems. Secondly， such an approximation should be “good” enough. Suppose we know the underlying problems can be described by certain functions， and sometimes with many measurements observations available， a natural question is how to model this problem approximately with functions. Precisely， how to find an approximation function 5(x) to approximate the underlying unknown function Suppose that the observations are function values ， if the approximation function s(x) is required to pass through the observed values at ， say ，then we call s(x) is an interpolation to f(x). If the difference between s(x) and f(x) is minimized under certain metric or measurement ， then s(x) is called an approximation to ， which is also often referred to as fitting in engineering. It is obvious that interpolation is a special case of approximation. Indeed， interpolation is the mathematical foundation of approximation. Sometimes， further constraints may be enforced for certain practical applications. For example， convexity and shape preserving is always a desired property in computer-aided design ， that is， the approximation function s(x) should have the similar convexity and shape with f(x). Such problems can be viewed as general fitting or approximation problems. The focus of this book is multivariate scattered data interpolation and approximation. By default， we believe the readers have already equipped with basic knowledge of one dimensional approximation. Otherwise， one can refer to textbooks on approximation theory and computer-aided design. For scattered data interpolation and approximation， the observations or measurements are generally irregular， not necessarily located on a mesh grid. For interpolation and approximation on mesh grids， one can use the tensor-product approaches. Besides， we also discuss the case when observations of measurements are linear functionals. In this book， we usually start with a physical model which requires scattered data approximation， hopefully， such models can stimulate readers’ interest and make the underlying motivation more accessible; and then we analyze the underlying problems mathematically in rigor; finally we discuss how to implement corresponding methods on computers. All introduced methods are programmable. Some of them are used in current popular software. Next， we shall introduce several examples. Through these examples， we shall illustrate where scattered data approximation problems come from and how to translate them to mathematical problems. 1.1.1　Problems from Applications In reservoir simulation， one would like to tell the oil distribution through the study of the permeability formation of oil. The distribution of permeability can be viewed as a function in M3. Certain samples can be obtained from dr ill-wells. The locations of drill-wells are usually scattered. Such a problem of reconstructing the permeability function is a scattered data interpolation problem. In weather forecast， a contour map of the temperature field can be obtained by interpolation of some data from meteorological observatories. Such observatories are scattered on the earth. This problem can also be viewed as a scattered data interpolation problem. If we further require the trends of temperature change， we may need further information like the velocity field of wind. This is a much more complicated scattered data approximation problem. In the problem of copying， imitation and archaeological restoration of paleontology， one often needs to reconstruct a new model through some measured values of the underlying object. Due to the complexity