Theory and Applications of Multilevel Approximation

真实的世界中的许多问题都是复杂的，错综复杂的。逼近理论是数学的一个分支，它研究如何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些问题。为了找到快速有效的数值算法，它往往是有益的，探讨所考虑的问题的多层结构。在本研究计画中，我们将探讨多层逼近与小波分析的理论与应用。我们将进一步发展多级逼近理论和谐波分析的相关领域，包括小波分析。然后，我们将考虑应用的一般理论的数值解偏微分方程和图像处理。在多层近似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。我们还将考虑流体力学中出现的问题。该研究的成功将推动科学技术的发展，从而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.