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

职业：用于复杂多尺度现象数值模拟的动态自适应小波算法

• 批准号: 0132664
• 负责人: Oleg Vasilyev
• 金额: $ 30.31万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-02-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0132664&HistoricalAwards=false
• 关键词: CAREER; Dynamically; Adaptive; Wavelet; Based

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.

在过去的十年中，您见证了小波分析的发展，这是一种全新的数学工具，该工具已被科学和工程学领域的各种领域迅速采用。在短时间内，它作为具有强大跨学科特征的明确数学主题达到了一定程度的成熟度。小波肯定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。但是，在不同地区和工程区域中产生的困难偏微分方程（PDE）的解决方案的解决方案非常有限。该职业建议的目标是多种多样的。第一个目标涉及基于小波的数值算法的艺术状态的发展。第二个涉及将方法应用于具有挑战性的工程问题，目前使用常规数值算法难以解决。该项目的最终目的是大规模科学计算中使用小波的教育和传播。更具体地说，我们的教育目标是（a）开发有关使用基于小波的数值算法来求解部分微分方程的全面研究生/上级本科课程，（b）开发和传播```'''''''''``'''``'''``'''``''''``'''''''小波代码，以及（c）编写教科书教科书``该项目将由三个部分组成：（1）算法开发，（2）高性能计算应用程序以及（3）课程和教科书开发。在项目的第一部分中，我们计划在这两个领域中改进和进一步开发动态自适应的第二代小波搭配（DASGWC）算法：（a）将方法扩展到复杂的几何形状，（b）（b）适应算法，以有效地在大量平行计算机上使用。同时，我们计划将进一步改进的第二代小波搭配算法应用于以下两类问题：（a）流体结构相互作用和（b）微尺度传热。选择这些问题是为了测试小波方法的强度，并评估它可以真正在传统的数值方法上大放异彩的程度。最后，我们将开发一门课程和一本有关高性能算法的教科书，特别着重于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，在该网站上，我们将发布一些基本的小波代码，以求解PDE，家庭作业和其他相关材料。我们认为，这样的服务对工程和科学领域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。