Acquisition of Computer Equipment for Development of Algorithms for Scientific Computing

该设备将用于支持数学科学的研究和研究生和本科教育。数学系在卡耐基梅隆大学有一个强大的程序在数值分析，吸引了一些最有才华的研究生在我们的博士课程。该设备将被主要研究者、访客、博士后研究员和他们的研究生使用，以加强正在进行的计算材料科学和多层次计算技术的研究项目。材料科学中的CMU程序在计算具有复杂细尺度现象的材料的性质方面具有应用价值。这些问题出现在冶金、磁学、优化设计等领域的研究中，这些领域经常观察到细微的变化。最近的数学进展表明，这些振荡是由能量因素控制的，通常可以用一个宏观的（缓慢变化的）量来表征，即杨氏测量。这些表征为这些材料及其性质的有效数值模拟和近似提供了基础。本提案考虑了解决此类问题的算法的开发，分析和测试。高效算法的研究计划，特别是多层算法，旨在使声学、电磁学、形状优化和无损检测的优化设计计算在计算上可行。这类问题的有效解决需要针对潜在问题的多个解决方案，以便分离出最优解决方案：如果没有正在研究的这类强大的计算算法，我们考虑的问题将是不切实际的。多层次计算技术是科学和工程领域中非常有效的求解方法。作者参与开发这些方法来解决各种问题，并取得了巨大的成功。研究了这些技术在非椭圆系统边界相关优化问题中的新进展，以及双曲进化问题的有效离散化和多层次时间推进技术。这些技术的应用包括电磁学、微波器件设计、声学降噪、无损检测和流体流动控制等方面的问题。

This equipment will be used to support research and graduate and undergraduate education in the mathematical sciences. The Department of Mathematics at Carnegie Mellon has a strong program in numerical analysis which attracts some of the most talented graduate students in our Ph.D. program. This equipment will be used by the principal investigators, visitors, postdoctoral fellows, and their graduate students to enhance ongoing research programs in computational material science and multilevel computational techniques. The CMU program in material science has application in computing the properties of materials that exhibit complex fine scale phenomena. These problems appear in the study of metallurgy, magnetism, optimal design, etc. where fine scale (microscopic) variations are frequently observed. Recent mathematical advances suggest that these oscillations are governed by energy considerations, and can often be characterized by a macroscopic (slowly varying) quantity, the Young measure. Such characterizations provide a basis for the efficient numerical modeling and approximation of these materials and their properties. The development, analysis, and testing of algorithms for the solution of such problems is considered in this proposal. The research program on efficient algorithms - especially multilevel algorithms- is aimed at making optimal design calculations in acoustics, electromagnetism, shape optimization, and nondestructive testing computationally feasible. Efficient solution of such problems requires multiple solutions for the underlying problem in order to isolate an optimal solution: the problems we consider would be impractical without powerful computation algorithms of the kind under investigation. Multilevel computational techniques are very effective solution methods in a variety of different fields of science and engineering. The authors have been involved in developing these methods for a variety of problems with great success. The proposed research deals with new developments of these techniques for boundary related optimization problems governed by non-elliptic systems, and for the efficient discretization and multilevel time marching techniques for hyperbolic evolution problems. Applications of these techniques include problems arising in electromag netism, design of microwave devices, acoustic noise reduction, non-destructive testing, and the control of fluid flows.