Theory and Applications of Multilevel Approximation

现实世界中的许多问题都是错综复杂的。逼近论是数学的一个分支，研究如何使用简单函数的组合来逼近复杂问题，并找到有效的数值算法来解决这类问题。为了找到快速有效的数值算法，探索所考虑问题的多层结构往往是有益的。在这个研究项目中，我们打算研究多层逼近和小波分析的理论和应用。我们将进一步发展多级逼近理论和包括小波分析在内的调和分析的相关领域。然后我们将考虑一般理论在偏微分方程数值解和图像处理中的应用。在多层逼近的许多可能的应用中，我们将研究图像处理和金融数学领域的问题。我们也会考虑流体力学所引起的问题。这项研究的成功将推动科学技术的发展，最终造福于人类社会。

Many problems in the real world are complex and complicated. Approximation Theory is a branch ofmathematics studying how to use combinations of simple functions to approximate complex problems and find efficient numerical algorithms to solve such problems. In order to find fast and efficient numerical algorithms, it is often beneficial to explore the multilevel structure of the problem in consideration. In this research project we propose to investigate theory and applications of multilevel approximation and wavelet analysis. We shall further develop multilevel approximation theory and related areas of harmonic analysis including wavelet analysis. Then we will consider applications of the general theory to numerical solutions of partial differential equations and image processing. Among many possible applications of multilevel approximation, we will investigate problems in the areas of image procession and financial mathematics. We will also consider problems arising from fluid mechanics. The success of the proposed research will advance the development of sciences and technology, and therefore ultimately benefit the society of human being.