Theory and Applications of Multilevel Approximation

点击翻译按钮获取中文摘要

Many problems in the real world are complex and complicated. Approximation Theory is a branch of mathematics 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.