Scientific Computing Research Environments for the Mathematical Sciences

当前，对数学专业本科生和研究生的培养，如果不接触应用数学和数值分析领域，不熟悉基于先进数学模型的计算机模拟现实世界，就很难被认为是完整的。为了加强这方面的培训，怀俄明大学数学系将购买一个计算机集群，用于几个领域的学生培训和研究，包括通过相互作用粒子近似模拟分形界面增长，不确定输入下飞机发动机音调噪声建模，开发近临界流体的三维流动模型，分析了大规模图和对流主导的反应扩散问题的并行算法。这些项目预计将推动数学/数值分析领域的一些高兴趣领域的最新技术，例如：谱元方法，复值线性系统的大规模解，域分解方法，随机过程和不确定性的分析，由Levy噪声驱动的这些过程的数值模拟和离散拉普拉斯算子的谱分析。它们将需要开发新的计算算法，包括顺序的和并行的；复杂流体物理研究的新解析/数值模型，包括近临界流体流动和多孔介质中的反应扩散流动随机分析和不确定性量化的新应用以及研究大型复杂网络问题的创新技术，例如互联网上出现的问题。这项研究的影响将跨越几个科学学科，与怀俄明大学在其2004-2009年学术计划中将跨学科计算科学确定为关键领域相一致。数学系与机械工程系、化学与石油工程系和计算机科学系之间现有的研究联系将因集群的存在而得到加强。其影响还延伸到具有重大社会和技术利益的问题。飞机噪声建模工具在当前大型运输机的设计中至关重要，避免了数百万美元的风洞和飞行实验的需要。模拟分形界面生长是控制尖端工艺过程（如化学气相沉积）的第一步。对临界流体流动的更好理解有望对提高采收率和先进材料制造领域产生影响。在GIS、万维网和DNA银行等领域，需要大量的图形算法来加速数字数据的处理。利用生物屏障来限制污染物在多孔介质中的流动，解决了一个主要的全国性环境问题。该集群的收购与开发计算科学和高性能计算课程的计划相结合。这些项目为课堂学习和论文研究提供了理想的选题。这些课程将极大地加强对数学学生使用现代分布式并行计算机的训练，并使他们接触到应用数学的最新研究领域。

The training of both undergraduate and graduate mathematics students can hardly be considered complete nowadays without their being exposed to the areas of applied mathematics and numerical analysis, and becoming familiar with computer simulations of the real world based on advanced mathematical models. In order to enhance training in this direction, the Department of Mathematics at the University of Wyoming will purchase a computer cluster that will be used for student training and research in several areas, including the simulation of fractal interface growth by interacting particles approximations, modeling of aircraft engine tonal noise under uncertain input, development of three-dimensional flow models for near-critical fluids, and analysis of parallel algorithms for massive graphs and convection-dominated reaction-diffusion problems. These projects are expected to advance the state of the art in a number of fields of high interest in the mathematical/numerical analysis community, such as: spectral element methods, large-scale solution of complex-valued linear systems, domain decomposition methods, analysis of stochastic processes and uncertainty, numerical simulation of such processes driven by Levy noise and spectral analysis of discrete Laplacian operators. They will entail the development of new computational algorithms, both sequential and parallel; new analytical/numerical models for the study of complex fluid physics including near-critical fluid flow and reaction-diffusion flow in porous media; new applications of stochastic analysis and uncertainty quantification; and innovative techniques for the study of large, complex network problems such as those arising on the Internet. The impact of this research will span several scientific disciplines, consistent with the University of Wyoming identification of interdisciplinary computational science as a critical area in its academic plan for 2004-2009. Existing research links between the Department of Mathematics and the Departments of Mechanical Engineering, Chemical and Petroleum Engineering and Computer Science will be enhanced by the presence of the cluster. The impacts also extend to problems of large social and technological interest. Aircraft noise modeling tools are of utmost importance in the design of current large transport aircraft and avoid the need for multi-million-dollar wind tunnel and flight experiments. Simulation of fractal interface growth is a first step toward the control of cutting-edge technological processes such as chemical vapor deposition. A better understanding of critical fluid flow is expected to have an impact in the areas of enhanced oil recovery and advanced materials manufacturing. Massive graph algorithms are needed to speed up processing of digital data in fields such as GIS, the World Wide Web and DNA banks. The use of bio-barriers to restrict flow of pollutants in porous media addresses a major nation-wide environmental concern. The acquisition of the cluster is coupled with a plan to develop a curriculum in computational science and high performance computing. The projects provide ideal topics for class learning and thesis research. They are expected to greatly enhance the training of mathematics students in the use of modern distributed parallel computers, as well as expose them to current areas of research in applied mathematics.