Acquisition of Computer Equipment for Development of Algorithms for Scientific Computing

这些设备将用于支持数学科学的研究以及研究生和本科生教育。 数学系在卡内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。程序. 这些设备将由主要研究人员，访客，博士后研究员及其研究生使用，以加强计算材料科学和多级计算技术的正在进行的研究计划。 CMU在材料科学中的程序在计算表现出复杂的精细尺度现象的材料的性质中具有应用。 这些问题出现在冶金学、磁学、优化设计等研究中，经常观察到细微尺度（微观）变化。 最近的数学进展表明，这些振荡是由能量考虑，并往往可以表征为宏观（缓慢变化）量，杨测度。 这种表征为这些材料及其性质的有效数值建模和近似提供了基础。 在本提案中考虑了这些问题的解决方案的算法的开发，分析和测试。 高效算法的研究计划-特别是多级算法-旨在使声学，电磁学，形状优化和无损检测计算可行的优化设计计算。 这些问题的有效解决方案需要多个解决方案的基本问题，以隔离的最佳解决方案：我们考虑的问题将是不切实际的，没有强大的计算算法的调查。 多层计算技术在科学和工程的各个不同领域中是非常有效的求解方法。 作者已经参与开发了这些方法的各种问题，取得了巨大的成功。 所提出的研究涉及这些技术的新发展的边界相关的优化问题所管辖的非椭圆系统，并为有效的离散化和多级时间推进技术的双曲演化问题。 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中出现的问题。

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.