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

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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.