Mathematical Sciences: Nonlinear Approximation

非线性近似及其在数值问题中的应用将在本项目中进行。由于这种近似方法的误差比传统的近似方法小得多，因此它变得越来越重要。重点将放在一般情况下小波分解的非线性方法上，特别注意在有理近似和样条近似中的应用。这项工作的基础是非线性近似在非线性偏微分方程理论以及图像和表面压缩问题中所起的作用。在非线性逼近中，用非线性流形代替近似函数的线性空间。具有自由结的样条、有理函数和某些类型的自适应逼近是研究最多的例子。多年来公认的非线性近似的优点是，它允许对具有奇点的函数进行更好的近似。这个领域的研究人员的观点并不是要确定一个给定的函数是否可以用从某个类中取来的函数来近似。相反，人们从类开始，要求在规定的误差范围内近似的所有函数的特征。最近的研究表明，用已知的、可识别的函数族（例如，通过它们的平均振荡来表征）来识别这些集合是可能的。许多最好的工作都局限于一维近似。本项目将考虑一种相对较新的多元逼近方法。传统的方法是将域分解成合适的小域，在小域上建立近似。一种更有前途的方法是使用小波的概念，将分划函数分成更易于管理的部分，使其更易于近似。

Work on nonlinear approximation and its applications to numerical problems will be undertaken in this project. This type of approximation has become increasingly important since it may give much smaller errors than traditional approximation methods. Emphasis will be placed on nonlinear methods in the general setting of wavelet decompositions with particular attention paid to applications to rational and spline approximations. Underlying this work is the role played by nonlinear approximations in the theory of nonlinear partial differential equations and in problems of image and surface compression. In nonlinear approximation, one replaces a linear space of approximating functions by a nonlinear manifold. Splines with free knots, rational functions and certain types of adaptive approximation are among the most studied examples. The advantage of nonlinear approximation, recognized for years, is that it allows for better approximation of functions with singularities. The point of view taken by researchers in this area is not to determine whether or not a given function can be approximated by functions taken from some class. Rather, one begins with the class and asks for a characterization of all functions which may be approximated within a prescribed error. Recent work shows that it is possible to identify these sets with known, identifiable, families of functions (characterized, for example, by their average oscillation). Much of the best work has been confined to one-dimesional approximation. This project will consider a relatively new approach to multivariate approximation. The traditional approach is one of breaking up the domain into suitable smaller domains on which approximation can be established. A more promising approach, using the wavelet concept, considers the partitioning function into more manageable parts which lend themselves to good approximation.