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.

小波分析是近十年来发展起来的一种全新的数学工具，它已迅速被科学和工程的各个领域所采用。在很短的时间内，它已经达到了一定程度的成熟，作为一个定义明确的数学学科，具有很强的跨学科性质。Wavelet无疑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，小波应用于解决物理和工程领域中出现的困难偏微分方程（PDE）一直非常有限。这个职业建议的目标是多方面的。第一个目标涉及的先进的基于小波的数值算法的艺术状态。第二个涉及的方法，具有挑战性的工程问题，目前很难解决使用传统的数值算法的应用。该项目的最终目的是在教育和传播小波在大规模科学计算中的使用。更具体地说，我们的教育目标是（a）开发一个综合性的研究生/高本科水平的课程，使用基于小波的数值算法求解偏微分方程，（B）开发和传播“易于遵循”的基于小波的代码，和（c）写一本教科书“小波和偏微分方程的数值解。“该项目将包括三个部分：（1）算法开发，（2）高性能计算应用，以及（3）课程和教科书开发。在该项目的第一部分，我们计划改进和进一步发展的动态自适应第二代小波配置（DASGWC）算法在这两个领域：（a）扩展到复杂的几何形状的方法和（B）适应算法的大规模并行计算机上的有效使用。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。这些问题的选择着眼于测试小波方法的强度和评估的程度，它可以真正闪耀在传统的数值方法。最后，我们将开发一门课程和一本高性能算法的教科书，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，这样的服务将是巨大的教育价值的研究生在工程和科学，因为很少有大学提供课程，科学计算使用小波。