Mathematical Sciences: Nonlinear Approximation

数学科学：非线性近似

• 批准号: 8922154
• 负责人: Ronald DeVore
• 金额: $ 4.43万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8922154&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Approximation

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.

该项目将在非线性近似方面进行工作及其在数值问题上的应用。这种类型的近似值已经变得越来越重要，因为它可能会产生比传统近似方法更小的错误。 在小波分解的一般环境中，将重点放在非线性方法上，并特别注意对有理和样条近似的应用。这项工作的基础是非线性近似值在非线性部分微分方程理论以及图像和表面压缩问题中所起的作用。 在非线性近似中，一个人用非线性歧管替换了近似函数的线性空间。具有自由结，合理功能和某些类型的自适应近似的花朵是研究最多的示例之一。多年来识别的非线性近似的优点是，它允许更好地近似具有奇异性的功能。 研究人员在该领域的观点不是要确定是否可以通过从某些类中获得的功能近似给定功能。相反，一个人从课程开始，并要求对所有功能的表征，这些功能可能在规定的错误中近似。最近的工作表明，可以通过已知的，可识别的功能系列（例如，通过平均振荡来表征）识别这些集合。最好的工作中的大部分工作都局限于一段时间的近似。该项目将考虑一种相对较新的多元近似方法。传统的方法是将域分解为可以建立近似值的合适较小域。使用小波概念，一种更有前途的方法将分区功能视为更易于管理的部分，使自己具有良好的近似值。