CAREER: Dynamically Adaptive Wavelet-Based Algorithms for Numerical Simulations of Complex Multi-Scale Phenomena

小波分析作为一种全新的数学工具，在过去的十年里得到了迅速的发展，并被科学和工程的各个领域所采用。在很短的时间内，它已经达到了一定的成熟程度，成为一门定义良好的数学学科，具有很强的跨学科特征。小波已经开始在许多领域产生影响，包括信号处理、数据和图像压缩。然而，小波在求解困难偏微分方程（PDE）中的应用在物理和工程的不同领域一直非常有限。这项职业规划的目标是多方面的。第一个目标涉及基于小波的数值算法的发展现状。第二个是将该方法应用于具有挑战性的工程问题，这些问题目前很难用传统的数值算法来解决。这个项目的最终目的是教育和传播小波在大规模科学计算中的应用。更具体地说，我们的教育目标是(a)开发一门关于使用基于小波的数值算法求解偏微分方程的综合研究生/本科水平课程，(b)开发和传播“易于理解”的基于小波的代码，以及(c)编写一本教科书“小波和偏微分方程的数值解”。该计划将由三个部分组成：(1)算法开发；(2)高性能计算应用；(3)课程和教科书开发。在项目的第一部分，我们计划在以下两个方面改进和进一步发展动态自适应第二代小波搭配（DASGWC）算法：(a)将该方法扩展到复杂几何形状；(b)适应该算法，以便在大规模并行计算机上有效使用。同时，我们计划将进一步改进的第二代小波配置算法应用于以下两类问题：(a)流固耦合问题和(b)微尺度传热问题。选择这些问题的目的是为了测试小波方法的强度，并评估它比传统数值方法真正发光的程度。最后，我们将开发一门关于高性能算法的课程和教科书，特别强调基于小波的自适应方法及其在现代计算环境中的实现。我们还将维护一个网站，在那里我们将发布一些用于求解偏微分方程的基本小波代码、家庭作业和其他相关材料。我们认为这种服务对工程和科学专业的研究生具有巨大的教育价值，因为很少有大学提供使用小波的科学计算课程。

The past decade has witnessed the development of wavelet analysis, a brand-new mathematical tool, which has been quickly adopted by diverse fields in science and engineering. In a brief period it has reached a certain level of maturity as a well-defined mathematical subject with a strong interdisciplinary character. Wavelets have certainly begun to make an impact in many areas, including signal processing, data and image compression. However, wavelet application to the solution of diffcult partial differential equations (PDE) arising in different areas of physics and engineering has been very limited. The objectives of this CAREER proposal are manifold. The first objective concerns the advancement of the state of the art of wavelet-based numerical algorithms. The second involves the application of the method to challenging engineering problems, which currently are difficult to solve using conventional numerical algorithms. The final aim of this project is in education and dissemination of the use of wavelets in large scale scientific computing. More specifically, our educational goals are (a) to develop a comprehensive graduate/upper undergraduate level course on the use of wavelet-based numerical algorithms for solving partial differential equations, (b) to develop and disseminate ``easy-to-follow" wavelet-based codes, and (c) to write a textbook ``Wavelets and Numerical Solution of PDEs." The project will consist of three parts: (1) algorithmic development, (2) high- performance computing applications, and (3) course and textbook development. In the first part of the project we plan to improve and further develop the dynamically adaptive second generation wavelet collocation (DASGWC) algorithm in these two areas: (a) extension of the method to complex geometries and (b) adaptation of the algorithm for efficient use on massively parallel computers. At the same time, we plan to apply the further improved second-generation wavelet collocation algorithm to the following two classes of problems: (a) fluid-structure interaction and (b) micro-scale heat transfer. These problems are chosen with an eye to testing the strength of the wavelet method and to evaluating the degree in which it can really shine over conventional numerical methods. Finally, we will develop a course and a textbook on high- performance algorithms with particular emphasis on wavelet-based adaptive methods and their implementation in modern computational environments. We will also maintain a web site where we will post some of the rudimentary wavelet codes for solving PDEs, homework assignments, and other relevant material. We feel that such a service would be of immense educational value to the graduate students in engineering and science, since very few universities offer courses in scientific computing using wavelets.